Gelinas was part of NHL history before he ever played his first game. Less than two months after the Los Angeles Kings selected him with the No. 7 pick in the 1988 NHL Draft, he was included as part of the package sent to the Edmonton Oilers in the Wayne Gretzky trade.

After starting the 1988-89 season with the Oilers and scoring his first NHL goal against the Minnesota North Stars on Oct. 17, 1988, Gelinas was returned to Hull of the Quebec Major Junior Hockey League.

However, Gelinas made the NHL for good as a 19-year-old in 1989-90, scoring 17 goals in 46 regular-season games as part of Edmonton's "Kid Line" with Joe Murphy (who was 22 years old) and Adam Graves (21 years old). Gelinas also had five points (two goals, three assists) in 20 Stanley Cup Playoff games to help the Oilers win their fifth championship in seven seasons.

The Quebec Nordiques, who had taken some heat for not selecting the Shawinigan, Quebec, native six years earlier (instead selecting forward Daniel Dore with the No. 5 pick), acquired Gelinas in a trade with the Oilers on June 20, 1993. But after he scored six goals in 31 games, the Nordiques placed him on waivers, and he was claimed by the Vancouver Canucks on Jan. 15, 1994.

Gelinas settled in as a depth forward with the Canucks and had nine points (five goals, four assists) in 24 playoff games to help them advance to Game 7 of the Cup Final before losing to the New York Rangers.

In 1995-96, Gelinas finally showed the scoring touch that had made him such a high draft pick. He scored 30 goals that season, then set NHL career highs in goals (35) and points (68) in 1996-97.

However, after he missed 16 games early in the 1997-98 season because of a knee injury, and the Canucks traded him to the Carolina Hurricanes on Jan. 3, 1998.

Carolina used Gelinas in a more defensive role, but he continued to produce offensively as well, including scoring 23 goals in 2001-02. One year later, he played a valuable role as a checking forward as the Hurricanes advanced to the Cup Final, losing to the Detroit Red Wings in five games.

Following the season, Gelinas signed with the Calgary Flames as a free agent on July 2, 2002. After getting 52 points (21 goals, 31 assists) in his first season in Calgary, Gelinas again found himself in the Stanley Cup Final after helping the Flames qualify for the playoffs for the first time in eight seasons. During the playoffs, Gelinas earned the nickname "The Eliminator" by becoming the first player in NHL history to score the game-winning goal in three series during the same playoffs. Two of those came in overtime, making him the first player to score three series-winning goals in OT during his career.

Despite Gelinas' heroics, the Flames came up short, losing in seven games to the Tampa Bay Lightning.

After playing in Switzerland during the 2004-05 season, Gelinas returned to the NHL in 2005-06 and spent two seasons with the Florida Panthers before playing his final NHL season with the Nashville Predators.

He retired after playing the 2008-09 season with Bern in Switzerland.

In 19 NHL seasons, Gelinas had 660 points (309 goals, 351 assists) in 1,273 regular-season games, and 56 points (23 goals, 33 assists) in 147 playoff games.

Following retirement, Gelinas had stints as director of player development with the Predators and as an assistant on Calgary's coaching staff.