Crandall was signed as an amateur free agent by the Braves in 1948. He was only 19 when he first played in a major league game with the 1949 Boston Braves. He appeared in 146 games for Boston in 1949-1950 before entering military service during the Korean War. When his two-year hitch was over in March 1953, the Braves departed Boston for Milwaukee, where – benefitting from a powerful offense featuring Hank Aaron, Eddie Mathews and Joe Adcock – they soon became both successful on the field and phenomenally popular off it. Crandall seized the regular catcher's job from veteran Walker Cooper in 1953 and held it for eight years, handling star Braves pitchers such as left-hander Warren Spahn and right-handers Lew Burdette and Bob Buhl. As a testament to Crandall's pitch calling skills, between 1953 and 1959, the Braves' pitching staff finished either first or second in the National League in team earned run average every year except 1955. Burdette credited Crandall for some of his success saying, "I never- well hardly ever- have to shake him off. He knows the job like no one else, and you can have faith in his judgment". On September 11, 1955, with the Braves trailing the Philadelphia Phillies 4-1 with two outs in the ninth inning, Crandall hit a dramatic grand slam home run to win the game. The Braves won National League pennants in 1957 and 1958, also finishing in second place five times between 1953 and 1960, and captured the 1957 World Series championship – the franchise's first title since 1914. Although he only batted .211 in the 1957 Series against the New York Yankees, Crandall had a solo home run for the Braves' last tally in a 5-0 win in the deciding Game 7.
Though rarely among the league leaders in offensive categories, he finished 10th in the 1958 Most Valuable Player Award voting after hitting .272, tying his best mark to that point, with career highs in doubles and walks; he also led the league in putouts, assists and fielding average, and won his first Gold Glove. In the 1958 World Series, again against the Yankees, he hit .240; he slugged another Game 7 solo homer, tying the score 2-2 in the 6th inning, though the Yankees went on to score four more runs to win the game and the Series.
Crandall averaged 125 games caught during the peak of his career, and he paid the price, missing most of the 1961 season due to shoulder trouble, which gave Joe Torre his opportunity to break in. While Crandall did come back to catch 90 games in 1962 – hitting a career-high .297, making his final National League All-Star squad and winning his last Gold Glove – he was soon replaced by Torre as the Braves' regular catcher. In 1962 he also moved ahead of Roy Campanella, setting the National League record for career fielding percentage; however, Johnny Roseboro would edge ahead of him before his career ended. After 1963, he was traded by the Braves to the San Francisco Giants in a seven-player deal; he played a backup role in his final three major league seasons with the Giants (1964), Pittsburgh Pirates (1965), and Cleveland Indians (1966).
In 1,573 games over 16 seasons, he finished with a batting average of .254 with 179 home runs; his 175 HRs in the National League trailed only Campanella (242), Gabby Hartnett (236) and Ernie Lombardi (190) among the league's catchers. His 1430 games caught in the National League trailed only Al López, Hartnett and Lombardi. He won four of the first five Gold Glove Awards given to a National League catcher, and tied another record by catching three no-hitters. He retired with the fourth most home runs by a National League catcher, and his career .404 slugging average also placed him among the league's top ten receivers. He ended his career among the major league career leaders in putouts (4th, 7352), total chances (8th, 8200) and fielding percentage (5th, .989) behind the plate, and ranked fourth in National League history in games caught. Crandall was a superb defensive player with a strong arm; he threw out 45.44% of the base runners who tried steal a base on him, ranking him 8th on the all-time list. He was selected as an All-Star eight times during his career: 1953–1956, 1958–1960, 1962. A powerful right-handed hitter, he topped the 20 home run mark three times. After having caught Jim Wilson's no-hitter on June 12, 1954, he added another pair in 1960 – by Burdette on August 18, and by Spahn a month later on September 16; amazingly, all three were against the Philadelphia Phillies. Richard Kendall of the Society for American Baseball Research devised an unscientific study that ranked Crandall as the fourth most dominating fielding catcher in major league history.
Crandall eventually turned to managing, and piloted two American League clubs, the Milwaukee Brewers (1972–75) and the Seattle Mariners (1983–84). In each case he was hired to try to right a losing team in mid-season, but he never enjoyed a winning campaign with either team and finished with a managing record of 364-469 (.437). In between those American League stints, he was a highly successful manager of the Los Angeles Dodgers' top farm club, the Albuquerque Dukes of the Triple-A Pacific Coast League, and also managed the Class A San Bernardino Stampede from 1995 to 1997. He remained in the Dodger organization as a special catching instructor well into his 60s. He also worked as a sports announcer with the Chicago White Sox radio team from 1985 through 1988 and with the Brewers from 1992-94.