Bryan Little scored twice and Ondrej Pavelec 1, at MTS Centre.
"I think we dodged some bullets and Pavelec was the best player on the ice tonight," said Wheeler. "We were fortunate that we got some good bounces. They didn't get some good bounces, and like I said, Pavelec bailed us out. That's two points. We're going to be happy with that and we're scoring some goals. That's a good thing."
Gabriel Landeskog lit the lamp for Colorado, which lost for the third time in four tries. Semyon Varlamov allowed all five goals on 25 shots to take the loss.
After a scoreless first period, Winnipeg jumped on top in the second with two early goals within a two-minute span.
The first came off the stick of Kane 1:45 into the middle stanza when he swept around the back of the goal before moving to the top of the right circle and sending a backhander towards the net that went through a screen and into the twine.
Then, at the 3:30 mark of the second, Andrew Ladd attempted a shot from the top of the right circle that was blocked by a defender and trickled towards the net. Colorado's Erik Johnson tried to sweep the puck away from the goal, but his clearing attempt went right to Little, who quickly wristed the disc past Varlamov.
Colorado cut its deficit in half while on the power play with 9:18 left in the second. Johnson carried the puck into the offensive zone on a rush and fed a pass to Landeskog at the left circle, where he wristed a shot past Pavelec's stick side.
Winnipeg, though, responded just 46 seconds later when Kane ripped a wrister from the high slot that went well high of the net, but Wellwood was there to grab the fortuitous bounce off the boards to the right of the net and stuff it home.
Ladd gave the Jets a 4-1 cushion with a power-play goal at the 5:40 mark of the third period with a one-timer from a tight angle to the left of the net that bounced off Pavelec's shoulder and under the crossbar.
The Avalanche were given a four-minute power play not long after when Kane was called for a roughing double minor, but Colorado's Kyle Quincey was whistled for a cross-checking penalty less than a minute into the man advantage to essentially nullify the power play.
Little's power-play goal with 4:19 left in regulation accounted for the final score.
"Overall, we just weren't good enough tonight in all areas of the game. I don't think we had everybody here on deck tonight," Avalanche head coach Joe Sacco said.
Wheeler recorded his first career four-point game...The Jets improved to 2-0 on an eight-game homestand...Dustin Byfuglien had two assists for Winnipeg...The Jets went 2-for-5 on the power play, while Colorado was 1-for-5 on the man advantage.