Kucherov scored on the power play at 19:09 to cut it to 2-1 before he tied the game 2-2 with his 13th goal of the season at 12:56 of the second period.
"Stick to the plan," Kucherov said. "We followed the plan. Just play to our system."
Ryan put the Oilers back in front at 15:55 after Warren Foegele rushed the net and Ryan knocked in the loose puck to make it 3-2.
"I don't think we gave up a ton, but the stuff that we did give up was quality, Grade-A, we can definitely clean that up," Ryan said. "Definitely not a matter of scoring more goals, it's a matter of keeping it out of our net. That's a recurring theme."
Tanner Jeannot tied it 3-3 at 3:54 of the third with a one-timer off a pass from Michael Eyssimont.
Evan Bouchard put Edmonton ahead 4-3 at 8:04 with a wrist shot from the high slot, but Stamkos tied it 4-4 with a power-play goal at 9:28, a one-timer from the right circle off a pass from Kucherov.
"I just like the way the guys dug in," Cooper said. "As the game went on, I thought we managed the puck well and then odd-man [rushes] started coming in our favor."
Edmonton was awarded a power play with 5:07 remaining when Austin Watson was called for slashing Bouchard but failed to score. The Oilers finished 0-for-5 with the man-advantage.
"I thought we could've gotten one on the power play, and obviously we could've gotten one at the end there to tie it up," Edmonton forward Zach Hyman said. "I thought we outplayed them for the most part; [we] just weren't ahead on the score sheet."
NOTES: Oilers forward Connor Brown had one shot on goal in 10:24 of ice time in his first game back after missing six with a lower-body injury. ... Edmonton center Connor McDavid was minus-2 with four shots on goal in 23:00, ending his five-game multipoint streak against the Lightning. ... The Oilers are 0-5-1 in games this season when they have had at least 40 shots on goal. ... Kucherov leads the NHL with eight first-period goals. .... Tampa Bay forward Brayden Point and defenseman Darren Raddysh each had two assists.