Elvis Merzlikins made 41 saves for Columbus (19-35-6), which is 3-1-2 in its past six.
The Blue Jackets were playing the second of a back to back after defeating the Edmonton Oilers 6-5 on Saturday.
"We just ran out of gas," Blue Jackets coach Brad Larsen said. "You could feel it in the third. They had some pressure, (and) we had a couple turnovers which didn't help.
"I thought the first half of the game we were real sharp. We needed more guys tonight to be on their game. We had a few guys who just weren't locked in and it cost us tonight. You're thrilled to build the lead because you need it, especially on the road, and you're on a back to back here, so we just ran out of gas and got a little bit sloppy, and it cost us."
Kaprizov cut it to 2-1 at 1:15 of the third. Ryan Hartman's wrist shot was redirected by Spurgeon, hit Columbus defenseman Andrew Peeke then deflected in off Kaprizov.
Kaprizov tied the game 2-2 at 6:00, scoring from the slot after Joel Eriksson Ek won a face-off during a 5-on-3 power play. It was his 16th power-play goal of the season, the most in Wild history.
"They knew how they were playing the game," Minnesota coach Dean Evason said. "Did we know we were going to get the result? No, but again there was no panic. We didn't know we were going to come back and win, obviously, but we had a good feeling about how we were playing the game. We were playing right and we were fortunate that we got rewarded."
Mathieu Olivier put Columbus ahead 1-0 at 16:20 of the first period. Eric Robinson intercepted an Alex Goligoski pass and sent the puck to Olivier, who beat Fleury down low.
Merzlikins denied Kaprizov on a breakaway and a rebound attempt from Hartman 11 seconds into the second period to keep the 1-0 lead.
"Obviously, after one or two saves you build confidence and it just keeps going and going and you feel better," Merzlikins said. "You feel that you are controlling the game, it's not the game controlling you. That's real important for a goaltender, I think. I was controlling the game."