The Sharks twice came from behind during the regulation 60 minutes and also rallied in the shootout to end a three-game losing streak. New Jersey had won three in a row.
Clowe scored with 3:39 to go in the second period to lift San Jose into a 2-2 tie, but David Clarkson put New Jersey back in front with 6:42 to play in the third.
It took a goal from Joe Pavelski with 33 seconds remaining in regulation to keep San Jose in the game.
Ilya Kovalchuk again put New Jersey in front by scoring in the opening round of the shootout, but Michal Handzus tied it once more in the second round and Clowe finished off the final comeback of the night by beating Johan Hedberg with the game-winner.
Antti Niemi, starting his second game since returning from a knee injury, had to stop just 15 shots to get the victory. Hedberg made 37 saves.
Swim Week Miami Beach 2014 [PHOTOS]