Darnell Nurse also scored, and Stuart Skinner made 20 saves for the Oilers, the Western Conference champions.
"The job is not done," Hyman said. "It's a great story, but you need to finish it. Everybody will forget if we don't finish it."
Game 7 of the best-of-7 series is at Amerant Bank Arena in Sunrise, Florida, on Monday (8 p.m. ET; ABC, ESPN+, SN, TVAS, CBC). It'll be the 18th Game 7 in Cup Final history but the first in 79 years to feature a team that has rallied from being down 3-0.
"It's been a hell of a story so far, but at the end of the day, we play to win and this is going to be the hardest game for us," Edmonton forward Leon Draisaitl said. "They're going to come out hard, they're going to play at home. We have to bring our game again. I'm just really proud of the way we gave ourselves a chance. That's what it's all . But by no means is this going to be easy, a walk in the park. This is going to be the hardest game of the series."
FLA@EDM SCF, Gm6: Foegele, Draisaitl combine for game's opening goal
The Oilers are trying to become the first team in 82 years to win the Stanley Cup after falling behind 3-0 in the Final. The 1941-42 Toronto Maple Leafs are the only team to successfully complete the comeback, defeating the Detroit Red Wings.
The Oilers are third team in NHL history and first since the Red Wings in 1945 to force Game 7 in the Final after falling behind 3-0.
Edmonton can become the fifth team in NHL history to win a best-of-7 series in any round after losing the first three games.
"Of course, when you are down 3-0 you are going to say you believe, but we truly, truly believe in each other and the ability to just win one game at a time," Oilers forward Ryan Nugent-Hopkins said. "That's all it comes down to. You don't need to worry winning two, or three, or four, you just win that game, and I think we all have that belief that if we play our best on any given night, we can win, so that's been our focus, and we put ourselves back in a good spot."
Aleksander Barkov scored, and Sergei Bobrovsky made 16 saves for the Panthers, the Eastern Conference champions, who were 0-for-3 on the power play, falling to 1-for-19 with two short-handed goals against in the series.
"Right now, if you walked into the room, there won't be a lot of happy ," Florida coach Paul Maurice said. "It doesn't have to be right tonight. You've suffered a defeat. You feel it. It hurts. You lick your wounds and we start building that back tomorrow. But who you are tonight means nothing to who you're going to be two days from now."
FLA@EDM SCF, Gm6: Barkov puts Panthers on the board with sweet goal
Foegele gave Edmonton a 1-0 lead at 7:27 of the first period. A turnover by the Panthers led to a rush chance for the Oilers, with Draisaitl's pass from the top of the right face-off circle finding Foegele at the bottom of the left circle for a quick one-timer.
The Oilers have scored first in each of their three straight wins. They've also led 2-0 in all three of those games.
Henrique gave them that two-goal lead at 46 seconds of the second period, scoring on a 2-on-1 with Mattias Janmark.
"Ever since we went down in the series we've talked playing from the right end, playing good in the defensive zone and getting rid of our mistakes and put the pressure on them," Janmark said. "I think we did that throughout the game. We played the right way, we got some looks, scored."
FLA@EDM SCF, Gm6: Henrique finishes Janmark's feed for 2-0 lead
The Panthers thought they got a goal back 10 seconds later when Barkov appeared to score, but it was overturned after the Oilers challenged for offside. Video review determined that Florida forward Sam Reinhart entered the zone before the puck five seconds before Barkov scored.
"Huge, huge moment in the game, obviously," Hyman said. "Just massive."
If the challenge went against the Oilers, it would have been a 2-1 game with the Panthers getting a power play for delay of game.
"I actually didn't think it was that close," Edmonton coach Kris Knoblauch said. "We were actually going to call it right away and we had a little more time to review it and were like, OK. The only hesitation was maybe there wasn't the right video. In my mind it was definitely offside."
Maurice said a linesman told him that the last video clip the officials saw on the review led them to ruling it as offside.
"There was no way I would've challenged that if it was reversed," Maurice said. "There was no way I thought you could conclusively say that was offside. … I'm not saying it's not offside. We'll get still frames, bring in the CIA, we'll figure it out. But in the 30 seconds that I would've made that call, I would not have challenged."
FLA@EDM SCF, Gm6: Oilers successfully challenge Panthers' goal
The Oilers, still up 2-0, went on to kill a Panthers power play that began at 10:57, and Skinner made a key right-pad save on Barkov’s shot from the right circle at 15:30.
Hyman then scored on a breakaway at 18:20 for a 3-0 lead.
Barkov made it 3-1 at 1:28 of the third period when he went around Foegele and tucked the puck in around Skinner's right pad.
"We didn't think we had it in the bag or anything like that," Hyman said. "We had to come out strong in the third and they actually got one, but it was a good buffer to have and we were able to manage the game after that."
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