Will Borgen's goal at 13:43 tied it 3-3, and Robertson scored the winner at 3:53 of overtime with a solo rush out of the defensive zone past Bruins forward Pavel Zacha after getting the puck from goalie Jonathan Quick.
J.T. Miller and Will Cuylle also scored for the Rangers (22-25-6), who ended a three-game losing streak and won for the second time in 10 games. Quick made 21 saves.
David Pastrnak had three assists, including his 900th career NHL point, and Elias Lindholm scored two goals, but the Bruins (30-20-3) lost for the second time when leading after two periods this season (24-0-2).
Morgan Geekie scored his fourth goal in as many games and Joonas Korpisalo made 24 saves.
Cuylle gave the Rangers a 1-0 lead at 9:45 of the first period, when he deflected Urho Vaakanainen's shot from the left point into the net.
Elias Lindholm made it 1-1 at 12:09. Pastrnak's attempted cross-ice pass to Hampus Lindholm in the left circle instead hit off Miller in the slot and went to Elias Lindholm, who put it in with a wrist shot.
Miller scored 26 seconds later with a one-timer from the right circle to give the Rangers a 2-1 lead.
Lindholm got his second of the night at 3:18 to make it 2-2. Charlie McAvoy intercepted Miller's clearing attempt through the middle of New York's defensive zone. He fed Pastrnak, whose shot in the high slot was blocked. Lindholm scored off the rebound from between the circles.
Geekie gave Boston a 3-2 lead at 8:52, seemingly surprising Quick with a fast, short side shot from the left circle close to the wall. Pastrnak got his 900th point on the play with the primary assist.
Borgen tied it with 6:17 left in the third period, when his shot from above the right circle appeared to deflect in off Bruins defenseman Andrew Peeke.
