Odom was nicknamed Blue Moon in grade-school by a classmate who thought Odom's round face resembled the moon. Odom led Ballard-Hudson High School in Macon, Georgia to two consecutive state championships while amassing a 42-2 record and pitching eight no-hitters. He signed with the Kansas City Athletics upon graduation.
Odom began his professional baseball career with the Birmingham Barons of the Southern League. After one season in Birmingham, he received a September call-up to the Athletics in 1964, and made his major league debut at just nineteen years old on September 5 at Municipal Stadium in Kansas City. Odom lasted just two innings against the New York Yankees, giving up a three-run home run in the first inning to Mickey Mantle and surrendered three more runs in the second before giving way to the bullpen.
Odom spent the entire 1965 season with the Lewiston Broncs of the Northwest League. His debut for the Broncs is locally legendary and was one of the most spectacular in baseball history. In the bottom of an inning with no outs, Odom came to the mound in relief. He not only struck out three in a row, he threw all strikes and left the mound to a rousing ovation (from an eye-witness account). For the season, he went 11-14 with a 4.27 earned run average, and led the league in games started (29) and innings pitched (198). He only made one appearance at the major league level all season, pitching one inning and allowing one earned run against the Washington Senators on September 22.
Odom split the 1966 season between Kansas City and the double A Mobile A's, going 5-5 with a 2.49 ERA at the major league level. He began the 1967 season in Kansas City, but was demoted in July with a 2-4 record and 5.15 ERA. He beat the Yankees 2-1 in his return. For the season, he went 3-8 with a 5.04 ERA.
Odom's arrival as a quality major league starter seemed to have coincided with his franchise's arrival in Oakland, California, as he improved to 16-10 with a 2.45 ERA the season after A's owner Charles O. Finley moved his franchise. He had a no-hitter against the Baltimore Orioles broken up by a Davey Johnson single with two out in the ninth on June 7, 1968. Baltimore scored a run in the first on three walks as Odom walked eight in the game. He finished third in the American League with 98 walks and tied Frank Bertaina for the league lead in wild pitches with seventeen. He was also named to his first All-Star Game, and pitched two scoreless innings.
Odom was simply dominant in the first half of the 1969 season, going 14-3 with a 2.41 ERA heading into the All-Star break. He also showed himself to be one of the league's better hitting pitchers as he went 3-for-3 with a home run and six runs batted in against the Seattle Pilots on May 4. He was named to his second consecutive All Star team, but was tagged for five runs (four earned) in just a third of an inning as the National League cruised to a 9-3 victory. His numbers tailed off considerably following the All-Star break, as he went 1-3 with a 4.09 ERA in the second half of the season.
After finishing second to the Minnesota Twins two years in a row in the newly realigned American League West, the A's finally won their division in 1971. Odom did not get the opportunity to pitch in the post-season as they were swept by the Orioles in the 1971 American League Championship Series. During the 1971 season, on July 11, Odom executed an unassisted double play. The A's would not see another pitcher execute an unassisted double play until April 27, 2017, when Kendall Graveman executed one with Odom in attendance at Angel Stadium.
Prior to the start of the 1972 season, Odom was shot twice by a burglar while trying to prevent a burglary at his neighbor's house. Odom went 15-6 with a 2.50 ERA in a rotation that also included Catfish Hunter, Ken Holtzman and Vida Blue as the A's cruised to their second consecutive division title in 1972.
In his first post-season start, Odom shutout the Detroit Tigers, holding them to just three hits in the second game of the 1972 American League Championship Series to give Oakland a 2-0 series lead. The Tigers came back to tie the series as Odom took the mound for game five of the ALCS. He held the Tigers to just one unearned run in five innings, and Vida Blue pitched the final four innings as the A's beat the Tigers 2-1.
Despite the fact that both pitchers pitched well, and their team was now headed to the World Series, a fight broke out between the two men after the game. After going 24-8 with a 1.82 ERA and winning the Cy Young Award in 1971, Vida Blue had a disappointing 1972 season in which he went 6-10. As a result, manager Dick Williams decided to use Blue as a reliever in the post-season.
Blue was unhappy with this decision, and made his feelings known. During the team's post-game celebration on October 12, Blue joked with Odom that he choked, and needed to be rescued by him. At some point, the joking turned serious, and nearly came to blows.
Oakland held a 2-0 game lead against Cincinnati's "Big Red Machine" in the 1972 World Series as Odom faced Jack Billingham in game three. Odom had allowed just one hit and two walks while striking out ten in six innings before giving up his first post-season earned run in the seventh inning. Unfortunately for Odom, Billingham was even more dominant, as he pitched eight scoreless innings to lead the Cincinnati Reds to a 1-0 victory.
Odom and Billingham were again embroiled in a pitchers' duel in game seven of the World Series, this time with Odom leading 1-0 before both pitchers gave way to their bullpens. The A's won the game 3-2, giving the franchise their first world championship since moving to Oakland. For the post-season, Odom was 2-1 with a 0.71 ERA and eighteen strikeouts in 25.1 innings pitched.
Though Odom fell to 5-12 with a 4.49 ERA in 1973, his A's returned to the post-season. Odom was used in relief by Dick Williams during the 1973 post-season. He made just one appearance in the 1973 American League Championship Series, pitching five innings and giving up just one earned run in their 6-0 loss to the Orioles in game two.
Odom made two appearances in the World Series against the New York Mets. In game two, he pitched two scoreless innings. Odom entered game four of the series with only one out, two men on and three runs already on the board in the first inning. Odom got Don Hahn to ground into a double play to end the inning. He left the game in the fourth without giving up a run, however, Darold Knowles allowed both base runners he inherited from Odom to score.
Odom was used in relief in the 1974 season, earning his only career save on August 30 against the Detroit Tigers. The A's once again faced the Baltimore Orioles in the ALCS, beating them in four games. Odom's only appearance came in their game one loss.
Just prior to the start of the World Series against the Los Angeles Dodgers, Odom and fellow A's reliever, Rollie Fingers got into a fight in the A's locker room after Odom made a comment about Fingers' wife. The incident lasted less than a minute, however, Fingers required six stitches on his head, and Odom sprained his ankle and had a noticeable limp.
Regardless of his injury, Odom pitched a scoreless ninth inning in game two, and earned the win in game five. Odom faced just one batter, Davey Lopes, in the fifth and final game of the World Series. Lopes grounded out to end the seventh inning with the score tied, 2-2. Joe Rudi led off the following inning with a home run to put Oakland up 3-2, and give Odom the win.
Odom made just seven appearances for the A's in 1975 before a May 20 trade sent him to the Cleveland Indians for Dick Bosman and Jim Perry. Odom was unhappy in Cleveland, and after just a month there, Odom was traded again, this time to the Atlanta Braves with Rob Belloir for Roric Harrison. It was the worst season of Odom's career, as he went just 2-9 with a 7.22 ERA for his three clubs.
Odom began the 1976 season playing minor league ball for the Atlanta Braves, and did not see any major league experience until a June 15 trade sent him to the Chicago White Sox for Pete Varney. In just his second start for the ChiSox on July 28, Odom combined with reliever Francisco Barrios to pitch a no-hitter against his former team, the Oakland A's. Odom pitched the first five innings, Barrios the last four, in what would be his last big-league victory.
Odom pitched six games for Oakland's triple A affiliate, the San Jose Missions in 1977 before retiring. He was asked to represent the Athletics at the 2008 First Year Player Draft, June 5 in Lake Buena Vista, Florida.
An excellent athlete, Odom was used quite often as a pinch-runner. He also wielded a strong bat (for a pitcher), hitting twelve home runs with a .195 career batting average. Odom had a 3-1 career record in the post-season with a 1.13 ERA and 27 strikeouts.
Odom was arrested twice in 1985. The first time was for selling cocaine to a co-worker. Then, during the trial, he was arrested a second time on December 11, for assault with a deadly weapon as he held his wife, Gayle, at gunpoint with a shotgun, holding police at bay for six hours. He ended up spending six weeks in alcohol rehabilitation and another 55 days in jail for the drug conviction.
Odom was inducted into the Georgia Sports Hall of Fame in 2004.