Vasilevskiy, recalled from Syracuse of the AHL Friday, made three more saves during the shootout. Ben Bishop, who has played in 31 games this season, was the backup one day after leaving practice early with an undisclosed issue.
“Many, many, many emotions,” said Vasilevskiy, whose wife gave birth to a son in Tampa Bay on Thursday.
Valtteri Filppula and Nikita Kucherov scored in regulation for the Lightning, who went 3-2-1 on a six-game homestand. Left wing Ondrej Palat was reunited on a line with Kucherov and Tyler Johnson after missing eight games with an ankle injury and had an assist.
``We got a point,'' Wild coach Mike Yeo said. ``If we won the shootout, we might be feeling pretty good about ourselves.''
Rob Martell worked his 1,000th and final game as an NHL referee, and was honoured in a pregame ceremony.
The Wild outshot Tampa Bay 15-3 in the third and pulled even at 2 on Spurgeon's backhand rebound with 2:39 remaining. Vasilevskiy made several strong stops during the period, including a pad save on Parise.
``That's when you need your goalie to play like he did,'' Filppula said. ``Huge game for him.''
Parise had a goal that he redirected during a power play overturned for incidental goalie contact after a challenge by Lightning coach Jon Cooper with 7:38 left in the third.
``Clearly a big part of the game,'' Cooper said.
``I had a feeling that was probably going to be the call, to be honest with you,'' Yeo said. ``That's the way it's going for us right now.''
The Wild are 6-5-6 on the road this season.
Kucherov, who has three goals and 12 points during an eight-game point streak, put the Lightning up 2-1 just over a minute into the second.
Filppula opened the scoring with 8:20 left in the first after eluding a sliding Spurgeon during a 2-on-1 and beating Devan Dubnyk with a wrist shot.
Dubnyk stopped 28 shots through overtime.
The Wild tied it at 1 with 4:21 left in the first when Coyle intercepted Vasilevskiy's clearing attempt from the behind the net and scored on a back-hander.
Minnesota was outshot 9-0 until Ryan Carter got the Wild's first shot 15 minutes into the game.
NOTES: Coyle assisted on Spurgeon's goal for his 100th NHL point. ... Matt Dumba set a Wild record for a defenceman with nine shots. ... Tampa Bay sent LW Jonathan Drouin, taken third overall in the 2013 draft, to Syracuse. Slowed by injuries, Drouin has two goals in 19 games this season. ... Minnesota recalled C Jordan Schroeder from Iowa of the AHL and put G Darcy Kuemper (concussion) on injured reserve. ... Kuemper has missed four straight games. ... Schroeder was scratched with Parise and LW Erik Haula (upper body) both in the lineup.