Hyman made it 2-0 with a power-play goal at 1:58 of the second period. Bouchard's shot from the point went in off Hyman's right leg for Edmonton's first 5-on-4 goal of the series.
"I score a lot of goals from there," Hyman said. "A lot of times it hits me and it goes in, [but] 'Bouchy' seems to hit me and most of the time it doesn't go in, it just hits me. It's nice that that one went in."
McDavid extended it to 3-0 at 5:00 by finding an opening between Bobrovsky and the left post with a shot from below the left face-off circle.
It was Edmonton's ninth straight goal in the series.
"Coming in on that side of the goal I've gone short side lots," McDavid said. "I would say most people know that I look there. [Bobrovsky] was standing. [Hyman] is always around the net, so I tried to put it there and it found a way in."
Tkachuk broke the Oilers' goal streak with his first of the series to cut it to 3-1 at 6:53.
But McDavid set up Perry for a power-play goal to push it to 4-1 at 11:54. McDavid went end-to-end, around Eetu Luostarinen at the offensive blue line and between Niko Mikkola and Dmitry Kulikov above the left hash marks before curling a pass to Perry in the slot for a tap-in.
"I watched him go through guys and then pass it over to me," Perry said. "I didn't even yell for it. He just saw me go to the net. That's just the type of player he is."
McDavid indicated that play was made from studying the Panthers and how they defend his breakouts.
"I'm in that position a lot going back for pucks, breakout, bringing it in the zone, so it's something that I look at a lot how certain guys are playing things," he said. "Mikkola has a really long reach and I just tried to work my way through there, and [Perry] did a great job of working it backdoor."
However, 14 seconds later it was Rodrigues scoring off a rebound of Montour's shot to cut it to 4-2 at 12:08.
Florida continued its comeback with Ekman-Larsson's goal at 4:04 of the third period to make it 4-3. Tkachuk, from below the goal line, found Ekman-Larsson cutting through the left circle and his one-timer went over Skinner's glove.
"It gave us a chance to come back in the game," Ekman-Larsson said. "We kept pushing, kept coming back and gave ourselves a chance."
The Panthers pulled Bobrovsky for the extra skater with 2:35 remaining, but McDavid scored into the empty net with 19 seconds left for the 5-3 final.
"I know a lot of guys counted us out," Edmonton coach Kris Knoblauch said. "We've been counted out a lot through the playoffs, regular season, whatever. But it doesn't faze the group in there. They have a lot of belief and a lot of just enjoying every extra day because we were counted out a long time ago and we're still here playing hockey in June. We are going back to Edmonton for Game 6 and there's a lot to smile about."
NOTES: McDavid has 42 points (eight goals, 34 assists) in the Stanley Cup Playoffs, the fourth most in a single postseason all-time. He's the third player in history to have at least 40 points in a single postseason, joining Wayne Gretzky (three times) and Mario Lemieux (once). … Edmonton's 13 goals when facing elimination (eight in Game 4, five in Game 5) are the most by a team since the 1942 Toronto Maple Leafs had 19. That Toronto team, which defeated the Detroit Red Wings in seven games, is the only team to win the Stanley Cup after falling behind 3-0 in the Final.