Mika Zibanejad, Jimmy Vesey, Adam McQuaid and Kevin Hayes all scored, Chris Kreider and Mats Zuccarello each had two assists for New York (24-23-8), which completed its season-long five-game homestand 2-2-1.
"It was a special birthday," said Georgiev, who made 23 of his saves on the penalty kill. "I tried not to concentrate on the birthday wishes because I knew I had an important game in front of me, and I'm very happy it ended the way it ended."
WATCH: [All Maple Leafs vs. Rangers highlights]
Kasperi Kapanen scored and Garret Sparks made 26 saves for Toronto (34-18-3), which had a season-high 56 shots on goal but couldn't extend its four-game winning streak and a six-game point streak (5-0-1).
The Maple Leafs hadn't lost in regulation since Jan. 20 (4-2 against the Arizona Coyotes).
"There was lots of chances on the power play, obviously, but I thought we had a lot of good players here tonight," Toronto coach Mike Babcock said. "It's not like I could go through it and say, 'This guy didn't play good.' That wasn't the case. We played good."
Georgiev's 55 saves are the most by a Rangers goalie in a game that ended in regulation since the NHL started tracking saves in 1955-56. It's also the most saves made by a Rangers goalie at the Garden in their history.
He fell four saves shy of matching Mike Richter's team record for most saves in a game; Richter made 59 saves in a 3-3 tie against the Vancouver Canucks on Jan. 31, 1991.