Spezza moved in slowly and outwaited Winnipeg goaltender Connor Hellebuyck before calmly sliding the disc past him into a wide open net.
Spezza had a good night, settting up the lone Maple Leafs goal in regulation as well, feeding Travis Dermott who scored from the left point.
Jack Campbell made 31 saves for Toronto and has gone 8-0-0 this season.
Andrew Copp scored the lone goal for the Jets.
Connor Hellebuyck made 36 saves for Winnipeg, who dropped to 22-14-2.
There was no scoring in the first period.
Travis Dermott opened the scoring 5:16 into the second when he converted a Jason Spezza feed from the blue line, beating Connor Hellebuyck with a short side wrister from the point.
The goal was his 2nd of the season.
Andrew Copp scored at 11:50, breaking in on a 2-on-1 break with Adam Lowry before snapping a pass from Lowry past Jack Campbell, beating him just inside the left post to get the Jets on the board.
The goal was his 12th of the season.
There was no scoring in the third period and the game went to overtime.
There was no scoring in OT and the game went to a shootout.
SHOTS-TORONTO 38 WINNIPEG 32
SHOOTOUT-TORONTO-Spezza (score), Matthews (miss), WINNIPEG-Connor (miss), Dubois (miss), Scheifele (miss)