It was the first multigoal game in the NHL for Cozens, who was tasked with head-to-head matchups against Oilers center Connor McDavid.
"It's a big confidence boost, for sure, to go against one of the top players in the world," Cozens said. "I was excited. I wanted to be up for the challenge and it's not just me, though. The whole team played a great game and my linemates and everyone. It was a big team effort to shut them down in the third (period) there."
Anders Bjork scored, and Dustin Tokarski made 33 saves for the Sabres (6-5-2), who were 0-4-1 in their previous five.
"We love him because he competes to win," Sabres coach Don Granato said of Tokarski, who got his first win since Oct. 16. "And it hasn't worked out as much as he'd like, but I don't think that's been his doing at all. He's done his part. I think the games that he hasn't won, we didn't score enough on our side, really. He's played great the whole season, and it's nice for that reason to see him win tonight."
Leon Draisaitl scored twice to bring his NHL-leading goal total to 14, and Ryan Nugent-Hopkins had two assists for the Oilers (10-3-0), who played their third game in four days. Stuart Skinner made 20 saves.
"We allowed them to score easy goals and we chased the game the whole night," Edmonton coach Dave Tippett said. "There was some effort there, but we just didn't play very smart."