Jon Merrill gave the Wild a 1-0 lead with a one-timer at 10:11 of the first period.
Talbot saved all 21 shots he faced in the first, but Cozens tied it 1-1 on a redirection at 8:05 of the second period after Kyle Okposo stole the puck from Talbot behind the net.
"It's hard to pinpoint one thing in that game," Talbot said. "We came out pretty flat obviously in the first period and it didn't really get a whole lot better from there. We were trying to make a push in the second period there, I turn the puck over, they get one and it's a 1-1 game.
"I just feel like I left too many secondary opportunities out there for them tonight. In a game that we weren't creating a ton ourselves, I've got to settle things down in our zone. Obviously, that one giveaway cost us tonight."
The Wild had five shots on goal but failed to score during a four-minute power play after Sabres defenseman Rasmus Dahlin was penalized for high sticking Joel Eriksson Ek at 8:25 of the second period.
"It's a very winnable game because of the work they put in and they deserved to win," Buffalo coach Don Granato said. " The game we felt was ours. There was an opportunity that we would have to win that game. ... They were dialed in, and they went after it."