Trailing 2-0 to start the final frame, the Penguins became their comeback when Chris Kunitz scored 13:42 into the period.
Sutter provided the equalizer just 51 seconds later and then capped the rally with an unassisted goal, boosting Pittsburgh to a season-high sixth straight win.
Marc-Andre Fleury needed to make only 14 stops to earn the win.
Tyler Seguin and Zdeno Chara notched first-period goals for the Bruins, whose stunning loss snapped a three-game winning streak.
Boston's Anton Khudobin turned aside 31 shots.