Nikolai Kulemin scored the shootout winner, and Vesa Toskala turned aside all three New York attempts to spoil Tortorella's Rangers debut.
A Stanley Cup winner with Tampa Bay in 2004, Tortorella replaced Tom Renney on Monday - a day after the Rangers' 3-2 overtime loss to Toronto at Madison Square Garden - and has had had one full practice with the team.
"I think this was definitely a step in the right direction," goalie Henrik Lundqvist said. "We looked faster and had more energy. He told us to go after them more, take more chances to get our offensive game going a little bit."
While that translated into more scoring chances, the Rangers ran into a hot goalie in Vesa Toskala. He stopped 31 shots and turned aside all three attempts in the shootout, improving his record to 6-0-4 since Jan. 19.
Toskala has responded particularly well since returning from a week off. He's turned aside 102 of 107 shots during the three games since and the Leafs have earned five of a possible six points.
"He's been playing outstanding," defenseman Ian White said.
"His last couple games he's pretty much single-handedly won us the games."
The shootout victory was just the third in 10 attempts for the Leafs this season. Nikolai Kulemin beat Lundqvist on Toronto's first attempt, and Toskala took over from there.
Niklas Hagman scored in regulation for Toronto.
Wade Redden replied for the Rangers.
New York currently holds down sixth spot in the Eastern Conference, but is just three points up on ninth-place Carolina and four ahead of 10th-place Pittsburgh.
Tortorella doesn't have much time to turn things around and admitted that he might not be able to do much about the players he thinks are out of shape.
"It's difficult during the season," he said. "We're 60-plus games in. I expect the team to be tired, all teams are tired at this time of year. I wouldn't say the whole team - I have some concerns about a few guys.
"It's a tough way to play. You have to be in shape."
The coach even called a timeout shortly after Hagman tied it in the third period because he thought some of his defenseman needed a rest.
It was a tedious opening 20 minutes as both teams fumbled the puck around and failed to generate any top-quality scoring chances.
The most interesting moment came a little over 6 minutes in when White and Markus Naslund dropped the gloves for a brief fight. It was Naslund's first fighting major since December 1993, when he squared off with Hartford's Randy Cunneyworth during his rookie season.
"It wasn't much of anything," White said.
Another long streak came to an end in the second period when Redden opened the scoring. The much-maligned defenseman beat Toskala with a long shot at 10:26, giving him his first goal in 58 games.
Redden was one of New York's prized offseason additions and will be one of the players Tortorella tries to spark over the final 20 games.
The game started to open up after the goal, with Toskala sprawling across his crease to stop a couple chances from Petr Prucha before Toronto's Lee Stempniak hit the post at the other end.
Hagman tied it at 10:21 of the third, making a quick move coming out of the corner and beating Lundqvist to the glove side.
"I'm not upset with the team," Tortorella said. "I just saw us sink a little bit in the third period. I just thought the play got taken to us in the third period.
"We want to finish (better in) games like that."
Notes: All six replacement coaches this season have lost their debuts ... Mark Bell didn't dress for the Rangers after being claimed on waivers earlier in the day from Toronto ... Boyd Devereaux was a healthy scratch for the Maple Leafs ... Toronto's John Mitchell and Jeff Finger left the game with upper-body injuries ... Tortorella is the 34th coach in Rangers history.
