Chris Mueller scored the game-winning goal 3 minutes, 18 seconds into overtime, giving the Texas Stars a 3-2 lead in the best-of-seven second round series.
Kevin Henderson had the primary assist after laying a drop pass to Mueller, who scored on the far side.
Texas can close out the series on Sunday at home. A win would set up a Western Conference championship series with the Toronto Marlies.
Cristopher Nilstorp picked up his fourth win of the playoffs and made five saves — four in overtime.
Nilstorp entered the game with 2:21 remaining in regulation after an injury to Jack Campbell.
Before the injury Campbell had played well, stopping 34 of 36 shots — including a second-period penalty shot.
Texas led 2-0 after first-period goals by Mike Hedden and Brendan Ranford, but Grand Rapids tied the game in the second period with a pair of goals.