Cousins' seventh goal of the season was his first in 20 games, since Feb. 12 against the Vegas Golden Knights. He was a healthy scratch for the 2-0 loss to the New York Islanders on Sunday, which ended Arizona's 0-3-1 road trip.
"I didn't like sitting out. You never want to see a game from the press box at this time of the year," Cousins said. "Going down the stretch, every game and every point is crucial for us, so it's nice to help contribute."
Arizona is 9-1-1 in its past 11 home games, with Kuemper starting each one. He is 12-5-2 in his past 19 starts.
"Like he's been all year, he's been our MVP, and you saw it tonight," Cousins said.
The Avalanche host the Golden Knights on Wednesday and the Coyotes on Friday. Arizona hasn't made the playoffs since 2012.
"Friday night is the most important Coyotes game in how many years, right?" coach Rick Tocchet said. "This is something we're looking forward to. This is a big, big game for us."