Elliott Eliason added 12 points and seven rebounds for Minnesota, while Hollins also chipped in six rebounds and four assists.
Marcos Knight led Middle Tennessee (27-7) with 19 points and nine rebounds.
Minnesota (22-14) will face Pac-12 regular-season champion Washington at Madison Square Garden on Tuesday. It will mark the Gophers' first appearance in the NIT semis since 2003.
Minnesota, which also won road games over La Salle and Miami in the tournament, handed Middle Tennessee only its second home loss of the season. The Blue Raiders had won 13 in a row at the Murphy Center since an 87-84 loss in double-overtime to Belmont on Nov. 20.
"Getting to New York to play in (Madison Square Garden) is special for anybody," Williams said. "I don't think any of us has played at the Garden, so it's special. We've been through some things this season with injuries and things like that. We have some tough young guys who have been growing up real fast."
The Blue Raiders narrowed the game to one possession in the final minute, but Minnesota converted 11 of 12 free throws, including its last eight in a row.
"It's learning from the past," said Hollins, who hit all eight of his free throws. "In the past we didn't close out games. We know what to do now to close out games. Today we focused and knocked our free throws down and sealed the game."
Williams, who scored Minnesota's first eight points, finished 8 of 11 from the field and 6 of 8 from the free throw line.
Minnesota's run to the semifinals follows a lackluster finish to the regular season in which the Gophers went 3-8 before the NIT, including a five-game losing streak in February.
"It's a great reward," said Minnesota coach Tubby Smith. "Our goal was to get to postseason play, get to the NCAA (tournament). They can feel good about themselves and not have to listen to naysayers say they're not any good or can't play or coach can't coach. That's what it does for them."
In a game featuring nine lead changes through the first 26 minutes, Minnesota took charge midway through the second half. The Gophers broke a stalemate by taking a 46-43 lead on an Austin Hollins 3-pointer with 13:55 remaining.
Minnesota led the rest of the way but struggled to put Middle Tennessee away in the final minutes. In the final 28.6 seconds, Middle Tennessee narrowed the lead to three twice and two once, but the Gophers made all eight free throws in that span to hold on.
"They were a tougher team in the gym today," said Knight, who scored 17 points for Middle Tennessee in the second half. "We didn't get the second-chance points like they did."