In his first game since returning from Sochi, Varlamov made 40 saves Friday and the Avalanche held on for a 4-2 win against the Phoenix Coyotes at Pepsi Center.
"I wasn't concerned about it," Roy said. "When I saw him after the Olympics, I saw a big smile on his face. I knew he was ready to bounce back. [Goalie coach] Francois [Allaire] had a couple meetings with him and he was well-prepared for this game. The key was giving him the night off [Wednesday], give him the time to feel really comfortable."
Varlamov made 18 saves in the third period, nine on three Coyotes power plays, and Gabriel Landeskog completed a three-point night by scoring a power-play goal with 3:11 remaining in regulation after Shane Doan's second power-play goal of the game with 6:02 to play cut the margin to 3-2.
"They came pretty hard in the third period," Varlamov said. "We sat back a little, but we scored two quick goals, which was really good. I think that was the key to the game. I didn't feel any pressure. I felt pretty good, actually. I think I played pretty solid hockey. The guys did a great job offensively. The goalie always likes it when the guys score lots of goals."
Varlamov practiced for the first time Monday after arriving from Russia and dressed as Jean-Sebastien Giguere's backup Wednesday, when the Avalanche lost 6-4 to the Los Angeles Kings.
"I think those days helped me," he said. "I was tired emotionally and it gave me some rest. So I got back into a normal routine, trained with Francois. I felt pretty good."
Colorado (38-17-5) is in third place in the Central Division with 81 points, three behind the second-place Chicago Blackhawks with one game in hand.
Phoenix (27-22-11) is fourth in the Pacific Division with 65 points but very much in the hunt for a berth in the Stanley Cup Playoffs with 22 games remaining.
"It comes down to the last  games," Doan said. "It always does. We have to figure out a way to get more points."
Paul Stastny had a goal and an assist for the Avalanche, and Nathan MacKinnon added two assists to stretch his point-scoring streak to 10 games, the longest by an NHL rookie since Chicago's Jonathan Toews had a 10-game streak from Oct. 10-31, 2007.
"It's nice, but I've been playing with some really good players," said MacKinnon, who has five goals and nine assists in the streak. "I feel lucky to be on a winning team like this."
Nate Guenin and Stastny scored goals early in the third period to give the Avalanche a 3-1 lead.
Guenin broke a 1-1 tie 48 seconds into the period with a shot from the right circle that beat goalie Thomas Greiss to the far side. It was Guenin's third goal in 80 career NHL games and his first since March 19, 2012, when he scored against the San Jose Sharks while playing for the Anaheim Ducks.
"It's about time," Guenin said. "Just good timing there. The puck slipped through and I kind of saw an opening. I got my new sticks and I got used to them yesterday, so I have to give them some credit. We wanted this one bad. We felt we let one slip away the other night. We really wanted these two points, especially in front of our home fans. We want this to be a tough place to play and the fans were rocking."
Stastny scored at 6:12 after Greiss made a save against MacKinnon, who used his speed to drive down left wing. The rebound came to Landeskog, who passed to Stastny in front for a shot into an open net.
"He was flying," Roy said of MacKinnon. "It was a great play by him going to the net and it was an outstanding play by Landy. It was a super pass to Paulie."
Doan, who had gone eight games without a point, scored his second goal after Avalanche defenseman Cory Sarich dived to knock the puck away from Martin Hanzal but was penalized for tripping. Keith Yandle fired a shot from just inside the blue line and Doan tipped it in.
The Avalanche responded with a power-play goal from Landeskog, who scored from the right circle off a pass from MacKinnon.
The teams traded first-period power-play goals, with the Coyotes striking first at 12:47 on a goal by Doan. Mikkel Boedker took a shot from the right circle that hit the far post and caromed to Doan in front for a chip shot into a half-open net.
PA Parenteau tied the game for the Avalanche at 15:38. Greiss made a big stop against Nick Holden driving to the net before Parenteau launched a long shot that hit the inside of the left post and bounced in.
The loss was the first for Greiss in five career games against the Avalanche. He finished with 29 saves.
"We were solid in a lot of areas again, but in the end a couple of mistakes cost us," Coyotes coach Dave Tippett said. "The power play late in the third really hurt us. It came down to a turnover, a penalty and a goal. The margin for error is very slim."