Ben Marshall, Travis Boyd and Nate Condon also scored for Minnesota (28-13-1), which advanced to the Frozen Four for the first time since 2005. Kent Patterson made 24 saves for the Gophers, who will play top-seeded Boston College in the Frozen Four in Tampa, Fla., from April 5-7.
Ferris State and Union (N.Y.) College will face each other in the other national semifinal.
"You've got to win," Minnesota coach Don Lucia said. "Nobody's going to give it to you. You've got to go play to win and tonight they did. And they have most of the year in that situation."
The Gophers held on to their three-goal lead, something they weren't able to do a week earlier against UND in the WCHA playoffs.
After winning the regular-season title, Minnesota faced North Dakota in the WCHA semifinals last weekend and held a 3-0 lead after dominating the first 30 minutes. But UND scored six straight times, including five in the third period, to beat the Gophers and went on to win the tournament with a 4-0 win against Denver the next day.
"We kind of wanted to get that last game out of our system against North Dakota," Matson said. "When we're facing North Dakota, if you can't get up for that game, you can't get up for any game. The bench was great all night. We were motivating guys on the bench and everyone was excited."
Haula, Matson and Boyd all scored in the second period for Minnesota, which outshot UND 12-2 in the second.
Danny Kristo and Mario Lamoreaux scored for North Dakota (26-13-3), which was 11-2 in NCAA regional games entering Sunday. Aaron Dell finished with 19 saves in goal for UND, which believed it could rally even when trailing by three late in the third.
After all, North Dakota had done it just a week earlier.
"I thought we needed to get one," UND coach Dave Hakstol said. "Whether or not we could have rattled them at all, I don't know. I don't know what we had for shots on goal in the third period, but I know we had some great opportunities."
Patterson stopped 14 shots in the third as North Dakota outshot the Gophers 15-4 in the final period. Patterson went on to win Most Outstanding Player in the region after stopping 55 of the 60 shots he faced in two games.