On the final play of the first half, teammate Colten Heagle picked off MarQueis Gray's risky pass and tossed the ball back to Williams as he was taken down. Williams raced 52 more yards to put the Bison (3-0) ahead 28-14 at the break.
Donnell Kirkwood rushed 10 times for 70 yards and two touchdowns for the Gophers (1-3), but once again, it was difficult to determine which team was from a BCS conference - and who was playing at home.
NDSU scored touchdowns on drives of 65, 83 and 71 yards in the first half before the highlight-reel play by Heagle and Williams as time expired.
Even when the Bison finally faced their first fourth down, with 6:25 left in the third quarter, Ryan Jastram kicked a 49-yard field goal with room to spare to stretch the lead to 31-14.
D.J. McNorton rushed 13 times for 92 yards for NDSU, and Brock Jensen completed 16 of 21 passes for 197 yards - plus 37 yards rushing and a score on six scrambles.
The Gophers, who gave the Bison three first downs via penalties on their first two possessions, rushed for 168 yards - including a 10-play, 95-yard march, all on the ground, to tie the game at 14. But the Bison played with more confidence, polish and discipline and seemed to have their game plan a step ahead of Minnesota's the entire game.
The Gophers rotated Gray and Shortell at quarterback, and they didn't have any rhythm when they needed it at the end. Shortell found tight end Eric Lair for a 20-yard touchdown pass on fourth-and-7 to bring the Gophers within 31-24 with 9:05 left, but they couldn't get any momentum after that.
This was the third time Minnesota had played North Dakota State since the Bison moved up from Division II, and this game was every bit as difficult for the Gophers as the meetings in 2006 and 2007. Minnesota needed a blocked field goal as time ran out to preserve a 10-9 victory the first year and lost 27-21 to NDSU the following season.
Bison fans accounted for perhaps one-fourth of the 48,000-plus crowd, and NDSU even brought its full band.
McNorton, who was held out of the previous game with a sprained ankle and had an extra week to recover because the Bison didn't play last Saturday, looked plenty agile cutting and planting in the first half before the Gophers began to bottle up the Bison ground game.
McNorton spun past cornerback Troy Stoudermire with a cunning juke on his way to a 58-yard gain late in the second quarter to set up his own score that gave NDSU a 21-14 lead with 24 seconds left in the half.
For the Gophers, struggling against an FCS foe in front of their own fans is nothing new, though it's not any less humbling and frustrating for a school with an athletics department budget about five times as big as NDSU's.
Minnesota lost to South Dakota last year, leading to the firing of head coach Tim Brewster a month later.
The Gophers barely beat South Dakota State in a three-point game the season before that. Two weeks ago, they lost at home to New Mexico State, an FBS team but frequently one of the weakest in the NCAA's top tier.