James Neal, who scored in regulation, missed a potential game-winner in the top of the third round and Marc-Andre Fleury extended the contest with a poke-check on Alex Pietrangelo.
Kunitz then scored the game-winner when he hesitated, faked Elliott to the ice and launched the puck into an empty net, securing the Penguins' seventh straight win entering the All-Star Game break.
Steve Sullivan also scored and Fleury made 32 saves while turning aside three more in the shootout.
Patrik Berglund scored twice and Elliott turned aside 37-of-39 shots for St. Louis.
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