The Penguins' penalty kill made Bonino's goal stand up by keeping the Sharks from scoring on a power play in the final minutes.
Bryan Rust and Conor Sheary also scored for the Penguins, who got 24 saves from Matt Murray.
San Jose's Martin Jones had 38 saves, but couldn't get in front of Bonino's shot, which was made possible by Kris Letang's pass from behind the net.
Tomas Hertl and Patrick Marleau scored for San Jose, which erased a 2-0 deficit to get back into the game.
Game 2 is Wednesday.
