Muhammad-Ali Abdur-Rahkman added 11 points and Duncan Robinson chipped in seven for the Wolverines, who won the Big Ten tournament championship. Moritz Wagner, who came in averaging 14.5 points, was held to five points.
Michigan (29-7) will face sixth-seeded Houston, a two-point winner over San Diego State, in the second round on Saturday.
Matthews recorded his fourth double-double this season.
Michael Oguine and Ahmaad Rorie scored 15 points apiece for Montana, which was making its first tournament appearance since 2013. The Grizzlies (26-8) were limited to 32.1 percent field-goal shooting, including 3 of 15 attempts from 3-point range.
The Grizzlies scored just two points in the first 11 1/2 minutes of the second half as the Wolverines gradually pulled away.
Wagner scored his first points with a dunk on Michigan's initial second-half possession. When Abdur-Rahkman hit a fadeaway jumper with 14:16 left, the Wolverines had their first double-digit advantage at 40-30.
Robinson's layup with just under 10 minutes to go made it 44-30. Rorie hit a 3-pointer after a timeout for Montana's first points in nearly 10 minutes.
The Grizzlies cut the deficit to nine with under six minutes left before Robinson knocked down another 3-pointer. Montana couldn't get closer than nine points the rest of the way.
Michigan survived a slow start to lead 31-28 at halftime.
The Wolverines went scoreless during the first four minutes while the Grizzlies took a 10-0 lead. Michigan steadily chipped away after breaking its scoring drought and took its first lead with four minutes left in the half on an Abdur-Rahkman trey.
Matthews scored six of his team's last nine points in the half, allowing the Wolverines to stay on top. He had 12 points by the break, while Oguine and Rorie combined for 21 of Montana's points.
--Field Level Media
Montana | Michigan | |
Michael Oguine 15 | Scoring | Charles Matthews 20 |
Michael Oguine 3 | Assists | Zavier Simpson 4 |
Bobby Moorehead 9 | Rebounds | Charles Matthews 11 |
Michael Oguine 4 | Free Throws Made | Charles Matthews 6 |
Sayeed Pridgett 2 | Steals | Muhammad-Ali Abdur-Rahkman 2 |
Jamar Akoh 1 | Blocks | Jon Teske 2 |