Blackwood, who had surgery on his ankle in June, was replaced at the start of the second period by Scott Wedgewood, who made 17 saves.
Yegor Sharangovich won it at 3:36 of overtime from the top of the right circle after Mark Jankowski tied it 1-1 at 14:45 of the third period.
Kieffer Bellows gave the Islanders a 1-0 lead at 3:23 of the first period, converting a one-timer off a pass from Mathew Barzal.
Ilya Sorokin got the start and made 11 saves for New York before being replaced at the start of the third period by Cory Schneider, who allowed two goals on five shots.
Devils center Jack Hughes did not return to the bench at the start of the third period after being cross-checked in the second period by Islanders forward Ross Johnston, who was assessed a penalty on the play.
"No real update (on Hughes)," Devils coach Lindy Ruff said. "He's feeling OK, we hope everything is OK, and we'll find out more tomorrow."
Asked if he was worried that Hughes could be sidelined, Ruff said, "not right now" and he was "hoping it's nothing serious and more precautionary."
