Schaefer ripped a one-timer from the right face-off circle off a short pass from Mathew Barzal, going short side past Karel Vejmelka.
Jonathan Drouin and Emil Heineman scored and David Rittich made 27 saves for the Islanders (10-6-2), who have won four games in a row, the past three in overtime.
JJ Peterka and Dylan Guenther scored and Vejmelka made 18 saves for the Mammoth (10-7-1), who have lost four of their past five games.
Heineman opened the scoring at 7:13 of the first period, scoring a 4-on-4 goal by swatting a backhand with his back to the net to give the Islanders a 1-0 lead.
Nick Schmaltz appeared to have tied the game for Utah at 11:41, but after video review it was determined that the puck never crossed the goal line and was ruled no goal.
Peterka scored at 13:02, tapping in a pass from Lawson Crouse to make it 1-1.
Guenther scored a 5-on-3 power play goal at 18:35, firing a one-timer past Rittich to give the Mammoth a 2-1 lead.
Drouin was credited with the goal that tied the game 2-2 at 13:44 of the third period when he kicked the puck from just outside the crease before it went off the stick of Utah defenseman Nate Schmidt and into the net. The play was initially ruled no goal for Drouin’s distinct kicking motion but then was allowed after video review.
