Wesley was selected by the Boston Bruins with the No. 3 pick in the 1987 NHL Draft after the defenseman scored 16 goals in each of his three full seasons with Portland of the Western Hockey League.

Wesley made the Bruins out of training camp in 1987 and finished with at least 33 points in each of his seven seasons with them. He scored an NHL career-high 19 goals in 1988-89, his second season, put up 58 points (14 goals, 44 assists) in 1993-94 and helped Boston advance to the Stanley Cup Final in 1988 and 1990, though it lost to the Edmonton Oilers each time.

Seeking help on defense, the Harford Whalers traded three first-round draft picks to the Bruins for Wesley on Aug. 26, 1994. His role changed with his new team, which moved after the 1996-97 season and became the Carolina Hurricanes. Wesley was relied on as a shutdown defenseman, penalty-killer and leader on the blue line.

Wesley never finished with more than 32 points in a season for the rest of his career, which lasted until 2008 and included seven games with the Toronto Maple Leafs in 2002-03 before he returned to the Hurricanes. He helped the Hurricanes advance to the Stanley Cup Final for the first time in their history in 2002, then was part of Carolina's championship team in 2006, playing all 25 playoff games in a run that ended with a Game 7 victory against the Oilers in the Cup Final.

After playing 78 games in 2007-08, Wesley retired at age 39 with 537 points (128 goals, 409 assists) in 1,457 NHL games, and 52 points (15 goals, 37 assists) in 169 playoff games. The Hurricanes retired his No. 2 on Feb. 17, 2009, prior to a game against the Bruins. Upon retiring, he worked in player development for the Hurricanes and St. Louis Blues and won the Cup for the second time in 2019 as director of defensive development with St. Louis.