Pat Maroon and Alex Barre-Boulet scored, and Brian Elliott made 28 saves for the Lightning (10-4-3), who are 8-1-2 in the past 11 games. Steven Stamkos scored the only goal in the shootout.
"I think as a collective group everyone's got to kind of step up and just have that next-man-up mentality," Cirelli said. "And I think tonight the guys did a good job of battling for 60 minutes."
Kevin Fiala had a goal and an assist, Alex Goligoski had two assists, and Kaapo Kahkonen made 20 saves for the Wild (11-6-1), who rallied from two goals down late in the third period to tie it.
"We went in and talked to them after the game, just said we hate doing what we keep doing but we love what we keep doing," Minnesota coach Dean Evason said. "We don't want to do it but we love the resilience. We don't quit. What we like is we play hard, but we play hard for each other.
"Clearly we want to play with a lead and we want to play right. But we have to learn how to do that, and we will. But in the meantime, we were pretty excited about how hard we pushed to come back for each other."
With the Wild playing with an extra attacker after Kahkonen was pulled, Fiala cut it to 4-3 at 17:12 with a slap shot that deflected off the stick of Lightning forward Pierre-Edouard Bellemare.
Joel Eriksson Ek then tied it 4-4 with 39 seconds remaining, jamming in a rebound in the crease.