Goncalves slapped in a loose puck in the crease for the winning goal off his own rebound after the Lightning killed off a tripping penalty to Nikita Kucherov at 5:30 of the extra period.
The best-of-7 series is tied 3-3. Game 7 will be at Benchmark International Arena in Tampa on Sunday.
Andrei Vasilevskiy made 30 saves for his eighth career Stanley Cup Playoff shutout for Tampa Bay, the No. 2 seed in the Atlantic Division, which evened the series for a third time.
Jakub Dobes made 32 saves for Montreal, the No. 3 seed in the Atlantic.
The tightly contested series has had four games go to overtime, including each of the first three. All six games have been decided by one goal.
Ivan Demidov’s penalty for goalie interference with 3:18 remaining in the third nearly proved costly to the Canadiens but Dobes made a right pad save on Brayden Point’s one-timer from the slot on the ensuing power play.
Jake Guentzel drove a shot off the left post on Tampa Bay’s third power play of the game at 6:22 of the third, moments after Kucherov hit the left post.
Dobes gave the Lightning an early scoring opportunity when he mishandled the puck behind the net. Yanni Gourde stripped the puck but fell as he came out front and his backhand completely missed a wide-open net at 2:25 of the first period.
Tampa Bay came close to scoring at 5:40 of the second when the puck slid between Dobes’ pads and was heading into the net before Phillip Danault cleared it off the goal line.
Vasilevskiy made a pair of saves on Ivan Demidov on a Montreal power play at 18:35 of the second period. He stuck out his left pad to deny the Canadiens forward’s one-timer before making a glove save on Demidov’s backhand off the rebound.
Cole Caufield hit the post in the first and the second, and Alexandre Texier put a shot off the left post at 2:58 of the third.