The win snapped a five-game losing streak in which the Rangers were outscored 20-5.
Gaborik, who hadn't scored more than once in a game since Nov. 30, completed his 10th career hat trick at 9:17 of the third period. He took a pass from Vinny Prospal, flew down left wing and ripped a shot past Avalanche goalie Craig Anderson for his 33rd goal of the season and a 3-0 lead.
The Avalanche, needing a win to climb within two points of Northwest Division-leading Vancouver, have lost three consecutive games in regulation for the first time this season.
"It's definitely huge, especially after losing five straight," said Gaborik, whose previous hat trick came in a Dec. 20, 2007 game against the Rangers while he was playing for Minnesota. "Coming here and getting a win here in Colorado, they're having a great year. It's important to get a good feeling about ourselves.
"It's a boost of energy for this club as well as for myself to finally get a win. We can feel good about ourselves."
Gaborik felt especially good for Johnson, who finished with 34 saves.
"He was unbelievable," Gaborik said. "He made key saves at key times. Congratulations to him to get his first win, to jump in there and be a huge part of this win."
Johnson faced 18 shots in the third period and lost his shutout with 7:21 to play when Cody McLeod put in the rebound of Kyle Quincey's point shot for his first goal in 30 games, since Nov. 23.
"It was disappointing, but I just wanted the win. That was the biggest thing," said Johnson, who made 21 saves in Saturday's 3-2 loss at Phoenix. "It happens. I would have liked to have had the shutout, but it was 3-0 at the time. That made it 3-1, so they still had a chance to win the game and that's what I was worried about, just getting that win.
"It's pretty exciting. You want to get that first one so you know you can win at this level. It's such a relief and obviously I'll remember this. Now I need to move on and work on getting No. 2.
"We all had to step up, including me, and start winning some games. I had to come in here and play really well and help the team as much as I could. Everybody battled hard and it was awesome to see. I'm just excited about the 'W.'"
Gaborik made it a 2-0 game at 4:57 of the second period after Prospal fed him with a behind-the-back pass. Gaborik skated to the front of the net and whipped the puck inside the left post.
"He made a great pass and Anderson was shifting to the other side," Gaborik said. "I pretty much had an open net. He's a great passer. When we're moving the puck and moving our feet, things will happen. We did that a few times."
The Rangers failed to score on a 5-on-3 power play a little more than a minute into the opening period, but took a 1-0 lead on Gaborik's 250th NHL goal at 13:53 for his 500th career point.
Gaborik took a pass from Sean Avery at center ice, skated into the Avalanche end and beat Anderson to the stick side, using Colorado defenseman Ryan Wilson as a screen.
"You feel better playing with the lead," Gaborik said. "We haven't gotten a lead in so many games. I think when you get a lead and get that first goal, you feel more comfortable and confident. So it was a huge moment."
The Avalanche has totaled just three goals in its losing streak after scoring 22 goals in a season-best six-game winning streak.
Colorado's top line of Wojtek Wolski, Paul Stastny and Chris Stewart combined for 15 shots against the Rangers but was a collective minus-5 while getting blanked for the third game in a row.
"I think we need to score," Stastny said. "When our line is doing well, it helps our team do a lot better. When our line is not scoring, it makes it a lot tougher for our team to get a win.
"We can't get frustrated. We have to simply our game and go from there. Sometimes you just need one of those goals to get going. Right now, we're battling to get one. The whole team is. We just need to keep our confidence and stay positive."
The win was the Rangers' first in Denver since taking a 6-3 decision at since-demolished McNichols Sports Arena on Dec. 31, 1998.
1 - 0 NYR
2 - 0 NYR