At the start of the 1995–96 season, San Jose traded Ozoliņš for Nordiques/Avalanche forward Owen Nolan. In 66 games during his first season, Ozoliņš scored 50 total points, with more than half on the power play. Ozoliņš won the Stanley Cup with the Avalanche in 1996. The following season Colorado won the Presidents' Trophy. After losing to the Detroit Red Wings in the 1997 conference finals, Ozoliņš finished third in votes for the Norris Trophy, awarded to the league's most outstanding defenceman. In the regular season, he scored a career-high 68 points, second in the league for defencemen after Brian Leetch. Ozoliņš scored his first career hat trick on December 6, 1999 against the Vancouver Canucks. 1999 was his final year in Denver, in which Ozoliņš made $4 million as Colorado's third-highest paid player.