Hagman came in very tight on Brodeur, whipping home the deciding goal while showering the goalie with a spray of ice shavings.
"I decided I was going to try something else and I felt pretty confident with it," Hagman said, "I didn't want to put snow in his face. That's why I felt a little bad. I didn't want to celebrate too much. I didn't want to be cocky."
After Hagman gave Toronto a 2-1 shootout lead, Vesa Toskala secured the win with a blocker save on Jamie Langenbrunner.
Tomas Kaberle kept Toronto alive, beating Brodeur in the third round after Patrik Elias scored for New Jersey.
In regulation, Matt Stajan scored twice, and Alex Steen, Jamal Mayers and Alexei Ponikarovsky added goals for the Maple Leafs, who bounced back from a 3-2 loss to Tampa Bay on Tuesday.
"We came in as a tired team and had 48 shots on goal," coach Ron Wilson said. "I thought we did an unbelievable job all night long."
Toskala earned his first win against New Jersey in four decisions, stopping 26 shots through overtime.
Toskala, nearly pulled earlier in the game, was stellar in overtime and the shootout.
"To be honest, I had the hook ready a few times," Wilson said.
"In the end, it worked out. He made some huge saves, especially the 4-on-3 penalty kill in the overtime. That seemed to give him some confidence."
The Devils got goals from David Clarkson, Dainius Zubrus, Jamie Langenbrunner, Colin White and Zach Parise but lost their third straight.
"We were very fortunate to get a point," Devils coach Brent Sutter said. "Two of the last three games, we haven't been very good. A lot of our players have to play a lot better. It was like our first game of the season and we hadn't had training camp. We weren't sharp at all."
Clarkson and Zubrus scored in the first period when Brodeur made 13 saves and the Devils led 2-0.
Clarkson got the offense started at 2:32 with a deflection of Mike Mottau's shot from the right point. Zubrus followed at 7:04, knocking a rebound past Vesa Toskala.
Stajan scored twice in the second period for the Maple Leafs, who stormed to a 3-2 lead.
Stajan deflected in Luke Schenn's shot from the right boards at 4:13. The assist was the first NHL point for Schenn, the No. 5 pick in this year's draft. Stajan connected again at 5:31, four seconds after Brodeur was penalized for high-sticking.
Steen gave Toronto the lead at 7:31.
Langenbrunner was credited with a deflection of Mottau's shot at 9:14 as Andy Greene got his third assist of the game.
White, netting only his 18th goal in a 531-game NHL career, put New Jersey back in front at 13:42.
Toronto again generated the bulk of the scoring chances, outshooting New Jersey 17-9 in the period.
In the third, Toronto reclaimed the lead on goals by Mayers and Ponikarovsky.
Persistent pressure by Toronto's fourth line paid off when Mayers tied it at 9:15. Ponikarovsky added a power-play goal, the Maple Leafs' third, at 10:25 for a 5-4 lead.
It didn't last long as Parise extended his goal streak to five only 52 seconds later, knocking home Paul Martin's rebound to tie it.
Toronto outshot New Jersey 18-9 in the third for a lopsided 48-25 advantage through regulation.
Schenn nearly cost the Leafs when he was penalized for playing with a broken stick, but Toskala bailed him out by making two point-blank stops on Langenbrunner.
New Jersey outshot Toronto 6-0 in overtime.
Notes: Langenbrunner was considered doubtful but returned after missing one game with a leg injury. ... Injuries to Brian Rolston (ankle) and Bobby Holik (broken pinkie) opened the door for RW Matt Halischuk to become the fourth player to make his NHL debut with the Devils this season. ... Toskala made his fifth straight start for Toronto. ... A big hit by Toronto D Jeff Finger late in the first period knocked Zubrus into the Leafs bench.
