The host Oilers, who lost the first three games in the best-of-seven series, pulled level at 3-3 to force a Game 7 showdown on Monday at Florida.
"Everyone in the locker room believes in us," Oilers left wing Zach Hyman said. "There's an unshakable belief in this bunch.
"No matter how dire the circumstances are, we always believe we have a chance."
No Canadian-based NHL club has won the Stanley Cup since the 1993 Montreal Canadiens. The Oilers have not taken the trophy since 1990, the last of their epic run of five titles in seven seasons.
But Edmonton could become only the second team in NHL history to win the Stanley Cup Final after dropping the first three games. Only the 1942 Toronto Maple Leafs have managed the feat, completing the historic fightback over Detroit. Not since 1945 has a team rallied from 0-3 down to force a Game 7.
The Panthers, who lost to Vegas in last year's Stanley Cup Final, seek their first NHL crown in their third appearance in the championship series, having also been swept by Colorado in 1996.
Edmonton's Warren Foegele, standing just outside the left goalpost, scored his third goal of the playoffs off a pinpoint pass from Leon Draisaitl just 7:27 into the contest to put the Oilers ahead.
The Oilers outshot Florida 11-2 in the first period, limiting the Panthers to their fewest shots on goal in any period all season.
Edmonton made it 2-0 only 46 seconds into the second period on a wrist shot from Adam Henrique, who netted his fourth goal of the post-season.
Florida's Aleksander Barkov appeared to have answered for the Panthers just 10 seconds later but the Oilers challenged the goal saying the visitors were offside on the play.
After a video review by league officials, the ruling was made that Florida was indeed offside on the play and the goal was disallowed, leaving Edmonton ahead 2-0.
Hyman scored his second goal of the final on a breakaway with 1:40 remaining in the second period to stretch Edmonton's lead to 3-0.
Ryan Nugent-Hopkins blocked a shot for the Oilers and the rebound bounced out of the Florida attack zone and into perfect position for Hyman, who broke free up the centre of the ice and flipped a backhand shot just under the right arm of Florida goaltender Sergei Bobrovsky.
Barkov answered for the Panthers just 88 seconds into the third period, beating Edmonton goalie Stuart Skinner to pull Florida within 3-1.
But the Panthers came no closer and after pulling Bobrovsky for an extra attacker, Edmonton's Ryan McLeod and Darnell Nurse scored into the empty net in the dying minutes to seal Florida's fate.
