Joel Eriksson Ek and Jared Spurgeon had goals in regulation for Minnesota, which headed into its All-Star break with two wins in a row.
Marc-Andre Fleury stopped 29 shots for the Wild, including a sprawling save to thwart Tage Thompson on a wraparound try with about eight minutes left in regulation. He also made a pair of stops about four minutes later with Buffalo buzzing in the Wild's end.
''You saw this morning that we're out of the playoffs, right? Not fun to see. A good win tonight and a good two points,'' Fleury said. ''At least now we've got seven days to see ourselves in the playoffs. Then when we get back, get back to business.''
Thompson scored in the shootout for Buffalo, but Jack Quinn was stopped by Fleury, and Gaudreau secured the win for Minnesota.
Wild coach Dean Evason hates shootouts, but will gladly take the result.
''We have a couple world-class goalies,'' he said, ''and those three guys have been pretty good for us at the top.''
Added Gaudreau: ''Zuccy and Kirill are spectacular.''
Minnesota has seven post-regulation wins this year, four via shootout. Since the start of the last season, the Wild lead the NHL with 23 victories in extra-session games.
Quinn and Zemgus Girgensons scored in regulation for Buffalo, which earned seven of eight points on a road trip through much of the Central Division that included victories in Dallas, St. Louis and Winnipeg. Ukko-Pekka Luukkonen made 33 saves for the Sabres, a league-best 12-2-2 on the road since Nov. 22.
''Not what we wanted tonight, but to get seven out of eight points is a pretty good job by that group,'' coach Don Granato said.
With the shootout loss, Buffalo has points in seven straight road contests for the first time since Feb. 8 to March 6, 2011.
''It's a long trip we're on now,'' Girgensons said. ''They're not easy. And especially at the end of it, it's hard to push it. I thought we did a pretty good job.''
Buffalo went up by one twice in the first two periods, only to see Minnesota quickly respond each time.
Early in the first, Quinn took advantage of a turnover by Minnesota and beat Fleury with a wrister from the slot, but Eriksson Ek converted a feed from Kaprizov 73 seconds later on the power play.
Left open at the back post, Girgensons converted a pass from Kyle Okposo midway through the second period. But less than three minutes later, Gaudreau found Spurgeon coming low in the left circle and delivered a perfect feed for an easy redirect.
Rasmus Dahlin thought he gave the Sabres a 3-2 lead, but his rebound crossed the goal line less than a second after the middle period expired.
''I thought it was in,'' Fleury said. ''The horn was a little bit delayed, right? It was after the puck went in, so I was like, `Ahh.' Extra emotions, right? Like, `Ahh.' Then, `Yay.' That was good.''
SABRES SANS TWO
Buffalo D Mattias Samuelsson took part in the morning skate, but missed his second straight game with a lower-body injury. C Dylan Cozens, fifth on the team with 17 goals and 43 points, missed his first game with an undisclosed injury. Both were game-time decisions. C Rasmus Asplund and D Kale Clague took their spots.
''We've played lots of hockey, and I don't want to make things worse at this point,'' Granato said. ''We're in a good position moving forward.''
Sabres: At home on Feb. 1 vs. Carolina.
Wild: At Arizona on Feb. 6.
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