After Eric Himelfarb tied the score for Grand Rapids with 49 seconds left in regulation while an extra attacker was on, a scramble behind the Manitoba net early in the extra session ended with Voce potting a shot off the back of Drew MacIntyre's leg.
The Griffins, who've now won their last five overtime decisions in the playoffs - including three straight against Manitoba - will have a chance to put away top-seeded Moose on Sunday in Game 6.
Jon Insana scored the Griffins' other goal in the second period on a feed from Voce.
Colby Genoway, Brad Moren and Jason Jaffray tallied for the Moose.
Jimmy Howard finished with 20 saves to improve to 4-0 all time in overtime games during the playoffs, while Manitoba's MacIntyre suffered his second overtime loss in five days despite making 43 stops.