Schultz took a pass at the blue-line from Jordan Eberle and stepped into a hard wrist shot that sailed past the glove of goalie Danny Taylor at 1:36.
The Edmonton Oilers' defenceman now has seven goals on the season and leads the AHL in scoring with 16 points.
Taylor Hall had the other goal for the Barons (6-5-1), who snapped a three-game winless skid.
Dustin Sylvester scored for the Heat (7-1-3) with 1:45 to go in regulation, forcing overtime.
Oklahoma's Yann Danis made 31 saves in the win, while Taylor stopped 32 shots.
There were no goals through two periods thanks to stellar goaltending on both sides, but Oklahoma City finally broke through at 7:46 of the third.
Hall began the play by centring the puck to Nugent-Hopkins in front. Though Taylor stopped him and Teemu Hartikainen with a sprawling save, Hall had creeped in and backhanded the puck over his glove.
Hall, playing just his fourth game for the Barons since returning from a shoulder surgery, picked up his second goal of the season.
The goal ended a goal less drought of 187 minutes two seconds, a new team record of futility, surpassing the old mark by more than two periods.
But the Heat tied the game late in regulation. Paul Byron, playing just his second game of the season, centred the puck to Ben Street in front of the net. Though Danis stopped his redirection Sylvester shot the rebound in from the side of the net, sending the sold out crowd of 7,046 fans into a frenzy before Schultz quieted the arena in overtime.