Cote, a native of Quebec City whose younger brother Alain also played in the NHL, was selected by the Hartford Whalers in the first round (No. 11) of the 1984 NHL Draft. However, the defenseman had his best offensive seasons with the Washington Capitals in the early 1990s.

The Whalers chose Cote after he scored 15 goals and 65 points in his second season with Quebec of the Quebec Major Junior Hockey League. He spent almost all of his age-18 season with the Whalers, finishing with three goals and 12 points in 67 games. But he played just two games with Hartford in 1985-86, splitting most of the season between Hull of the QMJHL (where he was the league's best defenseman despite playing just 26 games) and Binghamton of the American Hockey League.

He made the Whalers to stay in 1986-87 and played five seasons with Hartford before being traded to Washington on Sept. 8, 1991.

Cote had a booming shot and often got power-play time, but never scored more than eight goals in a season with Hartford. However, his offensive game blossomed after the trade. He scored 11 for the Capitals in 1991-92, then was one of three Washington defensemen (along with Kevin Hatcher and Al Iafrate) to break the 20-goal mark in 1992-93, scoring an NHL career-high 21 and finishing with 50 points. Cote followed that with 16 goals and an NHL career-high 51 points in 1993-94.

Although Cote never scored more than seven goals in a season again, he remained a valuable defenseman for the next decade with the Capitals, Toronto Maple Leafs, Chicago Blackhawks, Dallas Stars and Capitals again. He played his final 28 games in 1999-00 with the Stars and helped them advance to the Stanley Cup Final, but Dallas lost its bid to repeat as champion to the New Jersey Devils in six games.

Cote retired in October 2002 with 435 points (122 goals, 313 assists) in 1,171 NHL games, and 33 points (11 goals, 22 assists) in 102 Stanley Cup Playoff games.