Curt Brackenbury began his professional hockey career in 1972, signing a free agent contract with the Jersey Devils of the EHL after playing his junior hockey with the Sudbury Wolves of the Northern Ontario Hockey Association. He had a strong rookie campaign, potting 17 goals and 44 points in 68 games. In 1973-74, he moved on to the NAHL where he was told he needed to play a more aggressive style if he hoped to get noticed by the NHL. In 45 games with the Long Island Cougars, Brackenbury amassed 194 minutes in penalties while also collecting 28 points.
Brackenbury signed a contract with the Chicago Cougars of the fledgling WHA in 1973-74, appearing in four games and scoring a single point. He also spent a year and a half with the Minnesota Fighting Saints before moving on to the Quebec Nordiques, where he enjoyed the bulk of his success as a professional.
In 1976-77, Brackenbury put together a strong offensive season, scoring 16 goals and 29 points while racking up 146 minutes in penalties. It was also the most gratifying season from the team's perspective as well, as they won their only Avco Cup championship, turning back Bobby Hull and the Winnipeg Jets in a thrilling seven-game series.
When the Nordiques joined the NHL for the 1979-80 campaign, Brackenbury got his first taste of playing in the best hockey league in the world. In 63 games he scored six goals and 14 points. The following year he joined the Edmonton Oilers, being snatched in the Waiver Draft, but on a club made up primarily of young hotshot superstars, Brackenbury found it hard to even crack the starting lineup. By 1981-82, at the age of 30, Brackenbury was seldom used and he was let go. In 1982-83, he played in 44 games with Salt Lake of the CHL, hoping to make it back to the NHL. He did manage to dress for six games with the St. Louis Blues, but the writing was on the wall, and so he retired.
Brackenbury played in 141 NHL games, scoring nine times while counting 26 points.