After two scoreless rounds in the shootout, Sjostrom skated in and beat Marc- Andre Fleury on a forehand-to-backhand deke. Henrik Lundqvist then denied Sidney Crosby's backhand offering to seal the win.
Markus Naslund registered a goal and an assist for the Rangers, who have won six straight against the Penguins at home. Nikolai Zherdev's late third-period tally sent the game to overtime and Lundqvist made 27 saves.
"It was just amazing for us to come back," said Sjostrom.
Crosby and Darryl Sydor scored in regulation for Pittsburgh, which had its three-game winning streak halted. Fleury allowed two goals on 44 shots.
Naslund cut the Penguins' lead to 2-1 at the 5:09 mark of the third period. Jordan Staal went off for hooking and Pittsburgh cashed in on the power play seven seconds later, as Scott Gomez took a feed from Chris Drury before centering the puck to Naslund at the edge of the crease and the veteran forward beat Fleury through the five-hole.
"We've got to do the job," said Pittsburgh head coach Michel Therrien of protecting a lead. "We can't keep backing up."
In the waning moments of regulation, Lundqvist was pulled for an extra skater and Crosby's bid for the empty net bounced wide. Zherdev then took a backhand pass from Naslund, skated down the right wing and snapped a wrister over Fleury's glove, tying the score with nine seconds to go.
Sydor scored first for the Penguins, snapping a puck from the left point that pinballed off a pair of Rangers and beat Lundqvist 9:04 into the contest.
Crosby put Pittsburgh up 2-0 with 4:01 remaining in the second period. Alex Goligoski let go a slap shot from the top of the right circle that trickled through the pads of the Rangers netminder. Crosby, stationed at the side of the net and with a defender on his back, was able to bang home the loose puck.
Thirty-seven seconds later, the Rangers were awarded a five-on-three power play for a full two minutes, but failed to convert.