Trailing the series 3-1 Perry came to the rescue scoring his second of the game, jamming home a loose puck past Andrei Vasilevskiy from a scramble in the front of the Tampa net 13:15 into the second overtime.
Game six of the series will go Monday in Edmonton, one of two Canadian hub cities for the playoffs being used to limit travel and minimize COVID-19 risks.
The Stars still face an uphill climb with the 1942 Toronto Maple Leafs the only team in NHL history to come back from a 3-1 deficit in a final to hoist the Stanley Cup.
Facing elimination Dallas came out with purpose, Perry potting the only goal of a fast-paced opening period before Tampa would answer in the second with Ondrej Palat collecting his 11th of the playoffs.
The Lightning took the lead early in the third when Mikhail Sergachev's shot from the point ricocheted in off the post but with six minutes left in the period Joe Pavelski sent the contest to overtime stabbing a rebound past Vasilevskiy.
The goal was the 61st of the NHL playoffs for Pavelski, making him the all-time leading American scorer. (Reporting by Steve Keating in Toronto; Editing by Stephen Coates)