The Sharks came from two goals down in the third and Thomas Greiss gave up a lone score after coming on early to replace Antti Niemi.
Niemi allowed two goals on just six shots before Greiss came in and the Sharks faced a 3-1 deficit heading into the third period.
Clowe began the comeback with a backhander 9:11 into the period and Jamie McGinn tied it on a slap shot with 4:54 left in regulation.
Evgeni Malkin scored twice for the Penguins, but he had one of his team's three misses in the shootout.
Greiss wound up with 29 saves to get the win. Marc-Andre Fleury stopped 34-of-37 shots for the Penguins.