Brazzell attended Alice High School, where he helped start a run by the Coyotes as a playoff powerhouse through the 1990s. As a senior in the 1993 season, he caught 68 passes for 1,931 yards. At the time, it gave him a national record for most yards receiving in a single season. He also helped his team reach the Class 5A, Region IV finals.
Brazzell played for Blinn College before transferring to Angelo State University for the final two seasons (1996 and 1997) of his collegiate career. Despite playing only two seasons, he finished ninth on the Rams' all-time receiving list (1,565 receiving yards). As a senior, he earned second-team All-American and first team All-Lone Star Conference honors. He also was an All-Lone Star Conference selection as a junior.
Brazzell was selected in the sixth round (174th overall) of the 1998 NFL Draft by the New York Jets, because they coveted his track speed. He was waived on August 30 and later signed to the team's practice squad. He was released on September 5, 1999.
On September 6, 1999, he was claimed off waivers by the Dallas Cowboys. He made his NFL debut on special teams in the 41-35 season-opening victory against the Washington Redskins. He was cut on October 14 and signed to the practice squad. He was promoted to the active roster in week 14 and registered his first reception in a 20-10 victory against the Philadelphia Eagles, a 12-yard catch from quarterback Troy Aikman. The next week he finished with a season-high 62 yards on two receptions against his former team the Jets. He earned his first NFL start in the 27-10 playoff wildcard loss against the Minnesota Vikings.
In the 2000 season he played in 9 games, recording only two receptions for 12 yards, while playing mainly on special teams and finishing in a tie for seventh place with 10 special teams tackles. In two games, he led the team in special teams tackles. He missed the final two regular season games due to a shoulder injury. He was released on August 28, 2001.
Brazzell was signed to the Edmonton Eskimos practice roster on October 15. On November 4, he made his CFL debut in a 32-26 win against the Montreal Alouettes, a game in which he had three receptions for 76 yards. He also had one reception for nine yards in Edmonton's 34-16 Western Final loss to Calgary.
Brazzell played in 11 games in his second season, finishing fourth among Edmonton wide receivers with 30 receptions for 546 yards and three touchdowns. One of hia personal highlights was a 101-yard reception for a touchdown from Jason Maas in a 27-21 season-opening win against the Calgary Stampeders. He continued to make big plays in the post-season, registering 4 receptions for 79 yards in a 33-30 Western Final against Winnipeg. In the 25-16 Grey Cup loss against Montreal, he had three receptions for 65 yards.
Brazzell signed with the BC Lions as a free agent on March 25. He played in all 18 games for B.C. that season. He finished second on team, and sixth in the CFL, with 1,111 yards on 68 catches including six receiving touchdowns. He registered five 100-yard receiving games. He had perhaps his best game of the season in the Lions' 26-20 victory on September 20 against the Winnipeg Blue Bombers when seven receptions for 157 yards. He had the first multi-to game of his CFL career when he hauled in a pair of scores in the Lions' August 22 victory against the Hamilton Tiger-Cats. Brazzell had two catches for 11 yards in the Lions' 28-7 Eastern Semifinal loss in Toronto.
In 2004 Brazzell finished fourth on the BC Lions in receiving with 49 receptions for 906 yards and eight TDs in 16 games. He had three 100-yard games during that span. Brazzell was equally effective in the post-season for the Lions. He had three catches for 40 yards and one touchdown in B.C.'s 27-25 Western Final overtime victory against the Saskatchewan Roughriders. He also had a pair of passes in the Leo's 27-19 loss to the Toronto Argonauts in that year's Grey Cup championship.
In 2005 Brazzell signed with the Hamilton Tiger-Cats. During his brief stay in Hamilton, Brazzell had 22 catches for 252 yards and two TDs in six games. Brazzell had signed with the Ti-cats on the promise that he would only play the wide receiver position, but the Ti-cats coaching staff had Brazzell play slotback for them. This outraged Brazzell, as he saw himself as a deep threat receiver, something that playing slotback did not let him do. On August 9, 2005, Brazzell was traded to the Winnipeg Blue Bombers in exchange for Canadian receiver Kamau Peterson. The trade quickly showed to be one sided, with Brazzell hauling in 12 passes for 307 yards and two TDs in his first 3 games as a Bomber, surpassing his yard total and matching his TD total that he had posted in 6 games as a Ti-cat. Brazzell finished the season with 46 catches for 708 yards and 6 TDs.
2006 was a season of ups and downs for Brazzell. He started the season strong with a multiple touchdown game against the Edmonton Eskimos on July 1 and a couple more touchdowns in other games. By mid season his play started to flounder, with him being totally neutral during many games. At the end of the 2006 CFL season Brazzell picked his play up to finish the season with 43 catches for 604 yards and 7 touchdowns. Although Brazzell was scheduled to become a free agent on February 16, 2007, he chose to re-sign with the Blue Bombers.
Brazzell played 7 games for the Blue Bombers until he injured his ankle in September. In those games he had 17 receptions for 182 yards and one touchdown. He was released by the club on February 5, 2008 and is currently listed as retired on the CFL website.