Kirill Kaprizov scored two goals to extend his home point streak to 14 games, and Matt Boldy had two assists for Minnesota (22-13-2), which is 9-2-0 in its past 11 games.
"You always compare yourself to those teams that go all the way to the end," Wild captain Jared Spurgeon said. "They've been there the last little while, so to be able to put in an effort like that, it's awesome for us."
Filip Gustavsson made 34 saves before leaving with 2:03 remaining in the third period because of an illness. He was replaced by Marc-Andre Fleury, who did not face a shot.
"(He) skated off and [got sick]," Wild coach Dean Evason said. "Don't know anything else. Obviously, they're evaluating him. Don't know at this point. That was all I was told."
Brayden Point scored for Tampa Bay (24-12-1), which was playing the second of a back-to-back following a 4-1 win at the Chicago Blackhawks on Tuesday.
Brian Elliott, who started both games with Andrei Vasilevskiy out because of an illness, made 28 saves.
"It's his job," Lightning coach Jon Cooper said of Elliott. "That's why he's probably been in the League so long, because he can handle stuff like that. He's a true pro. He did everything for us tonight, we just didn't score enough."
Joel Eriksson Ek gave Minnesota a 1-0 lead at 14:30 of the first period. Addison's shot from the point went wide of net, but the rebound caromed off end boards to Eriksson Ek, who lost control of the puck as he tried to get it to his forehand but had it redirect in off the skate of Nick Perbix.