Toronto has lost nine straight at home (0-7-2), dating to a win over Edmonton on Feb. 6.
The Rangers captain moved in on goalie Jonas Gustavsson and wired a shot top corner in the third round to clinch it. Brad Richards also scored for New York in the shootout. Tim Connolly had Toronto's lone goal in the tiebreaker.
"I just like the way we found a way to win," Rangers coach John Tortorella said. "I don't think we're doing everything great. There's a lot of things to work on, but we found a way to win.
"I thought we were ugly as hell through a number of minutes, and with of the things we're doing, we have a lot of things to work on. But by killing penalties, blocking shots, we found a way to win a road game."
Marian Gaborik, with two goals, and Brandon Dubinsky scored in regulation for New York (47-21-7). Lundqvist made 22 stops, and Richards added two assists for the Rangers, who bounced back from a home loss to Buffalo on Friday.
"There was a lot of back and forth, a lot of momentum swings," Callahan said. "We stuck with things and kept going. We had a lot of big penalties at the end and were able to get two points out of it.
The Rangers, who lead Pittsburgh by three points in the Eastern Conference, are tied with Western-leading St. Louis with 101 points. The Rangers have one more win than the Blues.
"I don't want to talk about the Penguins. I'm tired of talking about them," Callahan said. "It's about us now."
"We're trying to win every game regardless of whether we get booed or we get cheered," said forward David Steckel, who had two assists. "We had that mentality tonight but fell a point short."
Both teams had chances to end it in overtime, including a shot from Toronto's Phil Kessel on a partial break that just went wide. Kadri was also stopped from point-blank range by Lundqvist.
The line of Crabb, Steckel and Connolly has been given the responsibility of checking opposing teams' top lines by Leafs coach Randy Carlyle. The trio has also chipped in offensively, as was the case on Saturday night.
"It's always nice when you can get a bit of offensive pressure on an offensive line," Crabb said. "When they're in their zone they're obviously not scoring, so the more time you can get in the offensive zone, the better defensively you're doing."
Carlyle, who took over for the fired Ron Wilson three weeks ago, has been impressed with Crabb.
"He seems to be around the puck," Carlyle said. "He's getting lots of minutes and their line is earning it, so it's a win-win situation for a young players who's starting to grow."
Gaborik scored his second of the night and 37th of the season at 4:19 of the third on a beautiful saucer pass from Richards. Gaborik took the feed in the slot and beat Gustavsson between the legs.
The Rangers' lead didn't last long as Connolly followed up a shot by Crabb to fire in his 12th goal 19 seconds later.
"(Richards and Gaborik) have to play a little defense, too, and they're both capable of doing that," Tortorella said. "The third goal is probably one of the greatest goals I've ever seen, as far as what Richie did at the blue line -- to stay onside, get the puck and then saucer one to Gabby.
"It was just one hell of a goal. But they can't get scored on twice by (Toronto's) fourth line."
New York took three consecutive minor penalties, but Toronto couldn't muster much during the power plays.
The Leafs' commitment to defense under new coach Randy Caryle produced just 16 shots for Toronto through two periods.
Crabb gave Toronto a 2-1 lead at 6:53 of the second after the Rangers failed to clear the zone. Maple Leafs captain Dion Phaneuf kept the puck in at the point and got it to Crabb in front for a shot that just slid over the line behind Lundqvist.
Dubinsky tied it at 15:29 off a faceoff in which the Leafs were outmuscled in front of their net. Dubinsky fished the puck out of a maze of bodies and buried it behind Gustavsson for his ninth of the season.
Gustavsson, who made his first start since March 11, said it's easy to get up for a game against a fellow Swede like Lundqvist.
"He's a good goalie, one of the best, and he's having a really good year and you always know you've got to be sharp," he said. "It's fun to play against him, but in the end you prepare yourself the same way."
Toronto opened the scoring 9:11 in after Gustavsson made a big stop on a backhand by Callahan, and Kessel and Franson broke the other way on a 2-on-1. Kessel gave a head fake before sliding a backhanded pass across to Franson for his fifth of the season.
Gaborik tied it on a shot off a faceoff in the Toronto zone that deflected off Phaneuf's stick and past Gustavsson at 12:19.
New York started a three-game road trip after a going 4-3 on a seven-game homestand. ... Rangers forward Mats Zuccarello fractured his left wrist in the first period of Friday's loss to Buffalo and is out indefinitely. Also scratched for New York were D Jeff Woywitka, D Anton Stralman, D Steve Eminger (undisclosed injury), C Chad Kolarik and LW John Scott. ... Sitting out for the Leafs were D Mike Komisarek, LW Clarke MacArthur (upper body), RW Mike Brown (thigh bruise), RW Joffrey Lupul (separated shoulder), RW Matt Frattin (lower body) and LW Nikolai Kulemin (broken finger). ... Toronto forward Ryan Hamilton of nearby Oshawa, Ontario, played his first home game for the Leafs. The captain of the AHL's Toronto Marlies made his NHL debut against New Jersey on Friday.