It was the first hat trick for the franchise since Eric Boulton scored a trio of goals for the Atlanta Thrashers on Dec. 18, 2010, and fans at MTS Centre threw dozens of hats on the ice in celebration.
Little scored two of the goals 32 seconds apart early in the third period and also had an assist. Blake Wheeler assisted on all three of Little's goals.
Chris Thorburn scored his first goal of the season, captain Andrew Ladd had a goal and an assist and Dustin Byfuglien also scored for Winnipeg (14-9-4).
Colorado (9-12-6) got a pair of goals from Jarome Iginla. Gabriel Landeskog assisted on both of them.
Rookie Michael Hutchinson made 20 saves for Winnipeg, which has only had an overtime loss to Boston in its last five games.
Semyon Varlamov stopped 25 of 30 shots for Colorado before he was replaced by Reto Berra after Little's third goal at 1:17 of the third period. Berra stopped six of the seven shots he faced.
Colorado came off a 4-3 overtime loss to Calgary on Thursday.
Thorburn netted his goal off a rebound from defenceman Paul Postma's point shot, sending the puck over Varlamov at 4:31 of the first period.
Winnipeg has now scored first in its past eight games, and in 18 of its 27 games. Its record in those games is 11-4-3.
The Avalanche didn't get their first shot on goal until almost nine minutes into the period.
Their second shot resulted in Jerome Iginla's fifth goal of the season. It went over the glove of Hutchinson, who may have been partially screened by defenceman Mark Stuart, at 9:19.
Little's 2-1 go-ahead goal was reviewed, with a ruling that his shot from a sharp angle was caught in Varlamov's glove behind the goal-line at 12:29.
Winnipeg outshot the Avs 16-4 in the first.
The second period featured missed opportunities by both teams until Ladd scored his ninth of the season with 45 second left.
Avalanche centre Nathan MacKinnon had a breakaway, but fired his shot over the net.
The Jets had a three-on-one, but Evander Kane's shot went wide of the net.
Ladd tipped in defenceman Ben Chiarot's shot for the goal at 19:15, giving Chiarot his first NHL point in his second career game. He was called up from the AHL St. John's IceCaps earlier in the week.
Winnipeg had widened the shot gap 26-9.
Little scored his 10th and 11th goals of the season 32 seconds apart to open the third period.
His first goal beat Varlamov 45 seconds into the period and his second came after a scramble around the net. Little was turning around just outside the crease, saw the loose puck and sent it into the net at 1:17 for the 5-1 lead.
Tempers flared with just under 14 minutes left in the game, with an early tripping call on Winnipeg and roughing penalties from the dustups giving Colorado a two-man advantage for 13 seconds.
Little had a chance for a fourth goal into an empty net midway through the period, but Avs centre Ryan O'Reilly was back and blocked the shot.
Iginla scored his sixth of the season on the power play at 10:24 after Zach Redmond sent a pass across the crease as he was falling and Iginla popped the puck over Hutchinson.
Byfuglien extended his goal-scoring streak to three games with an unassisted blast from the point that beat Berra on the glove side at 13:20.
Winnipeg outshot the Avs 37-22.
Injuries to three Winnipeg defencemen, including the top pair of Tobias Enstrom and Zach Bogosian, meant Byfuglien was back playing defence instead of the wing.
Grant Clitsome (upper body) is expected to miss about two weeks, Enstrom (lower body) up to three weeks and Bogosian (foot) might be gone four to six weeks.
Winnipeg hosts Anaheim on Sunday afternoon to finish a three-game homestand. Colorado returns home and doesn't play again until Nashville visits on Tuesday.