The Wolverines, seeded fourth in the South Region, had only a six-point lead midway through the second half before disposing of the regular-season and tournament champion of the Summit League.
Robinson went 8-of-9 from the field and added six rebounds, pushing Michigan into a third-round meeting with Virginia Commonwealth, a Final Four participant two years ago which crushed Akron 88-42.
Michigan hit half its field goal attempts, turned the ball over just nine times and closed the game on a 22-13 run. Mitch McGary added 13 points for the Wolverines (27-7).
South Dakota State built an early advantage, but gave up the lead for good with 5:44 to go in the first half.
The Jackrabbits (25-10) were led by 20 points from Brayden Carlson.