Garrison's game-winner came on a breakaway at 2:17 of the extra session, which featured end-to-end action in the NHL's first 3-on-3 OT of the regular season.
The NHL moved overtime from 4-on-4 to the new setup this season.
It was the first NHL game for Philadelphia coach Dave Hakstol, who spent the past 11 seasons coaching at the University of North Dakota. Seven of his teams at North Dakota reached the NCAA Frozen Four.
The Lightning also got a goal from Ryan Callahan.