Brown tied it 3-3 when he chipped the puck over Anthony Stolarz's glove off a centering pass from Arseny Gritsyuk as he crashed the net.
Gritsyuk had a goal and an assist, and Nico Hischier had two assists for the Devils (31-29-2), who have won three straight after five losses in a row. Jacob Markstrom made 24 saves in his second straight start after a 5-1 win against the Florida Panthers on Tuesday.
Paul Cotter and Jesper Bratt scored in the shootout and Markstrom stopped William Nylander and Auston Matthews to clinch the win.
The Devils were without defenseman Brett Pesce, who sustained a lower-body injury against the Panthers on Tuesday.
Nylander, Matias Maccelli, and Matthew Knies each scored for the Maple Leafs (27-24-11), who have lost five straight (0-3-2) and 11 of their past 14 (3-8-3). Stolarz made 44 saves.
Forwards Scott Laughton and Bobby McMann and defenseman Oliver Ekman-Larsson did not play for roster management purposes.
Maccelli gave Toronto a 1-0 lead on a wrist shot from the slot at 5:26 of the first period.
Timo Meier scored his third goal in five games for New Jersey, making it 1-1 on a snap shot from the left hashmark at 7:55.
Nylander scored on a deflection off his lower back at the left post while on the power play at 3:20 of the second period to give the Maple Leafs a 2-1 lead.
Gritsyuk tied it 2-2 with his second goal in as many games with a one-timer just after a New Jersey penalty had expired at 6:06 of the second.
Knies gave the Maple Leafs a 3-2 lead when he batted the puck into the net from the right post at 13:50 of the third period.
