The Penguins held a 2-1 lead midway through the final period, but Pittsburgh's Matt Cooke drew a 5-minute major and a game misconduct on a elbow delivered to Ryan McDonagh. Matt Niskanen was whistled for a high-sticking penalty 2 1/2 minutes later, putting Pittsburgh two men down.
The Rangers responded with goals by Gaborik and Callahan only 11 seconds apart, and New York went on to win its fourth straight and strengthen its hold on the seventh seed in the Eastern Conference.
Callahan added a pair of assists, Brandon Dubinsky contributed a goal and two helpers and Henrik Lundqvist turned aside 38-of-40 shots for the Rangers.
Artem Anisimov notched a goal and an assist and Derek Stepan also found the net for the Rangers.
Jordan Staal and Chris Kunitz tallied and Marc-Andre Fleury made 22 saves for Pittsburgh.