The Dutchmen, who had not played since March 2, improve to 18-12-5 in taking the weekend opener at Messa Rink, while the fifth-seeded Big Green, fresh off a three-game victory over Harvard last week in Hanover, drops to 15-13-5 with the loss.
The Crimson also took game one of that first round series at Thompson Arena before the home team rattled off two straight to advance to the league quarterfinals.
Daniel Carr scored the first of his two goals on the night in the opening frame and added another midway through the third to help the Dutchmen take a 1-0 series lead in the first-ever postseason meeting between the two teams.
Sophomore Tyler Sikura evened the game at 1-1 at 15:12 of the second period, but Union's Daniel Ciampini answered with the go-ahead goal 2:47 later to put the home team back up by one again. Ciampini's marker would stand as the eventual game-winner with the Big Green unable find the back of the net the rest of the way.
"It's a game of winning battles," head coach Bob Gaudet said. "I thought we had a good second period and were working really hard to make a couple of good plays to get back in it.
"That's a very good team over there (Union), but if we can come out tomorrow night and work hard, I like our chances," Gaudet added.
Dutchmen netminder Troy Grosenick finished the night with 31 saves, while Dartmouth freshman Charles Grant needed to make 37 stops in his third straight start.
Neither team managed a power-play goal as Dartmouth was 0-for-3 on the night, while Union failed to capitalize on its four chances, including a five-minute power play in the third.
Game two will be Saturday night at 7 p.m.