Gary Bell Jr. led No. 7 seed Gonzaga (26-7) with 18 points. The Bulldogs erased a 10-point deficit to tie it at 61 on a 3-pointer by Elias Harris with 4:05 remaining, but Sullinger knocked down two pretty hook shots late while the Zags went cold.
The steady play of Craft and some gritty defense in the final minutes helped Ohio State avoid becoming the third No. 2 seed to lose in less than 24 hours. Fellow heavyweights Duke and Missouri fell to upstarts on Friday from one-bid leagues.
Yet the Buckeyes knew the Bulldogs are hardly scrappy underdogs. They entered the game with 17 NCAA victories over the last 14 seasons, the same as the Buckeyes.
The 18th will have to wait until next year at least, though Gonzaga again showed the mettle that's made it perhaps the most successful mid-major program in the country.
The Bulldogs appeared to be out of it after a putback by Thomas gave the Buckeyes a 58-48 lead. Gonzaga responded with a 13-3 run capped by a 3-pointer from the corner by Harris.
Then Sullinger, who still thinks about last year's loss to Kentucky in the regional semifinals, took over. He backed down Gonzaga center Robert Sacre and hit a soft little hook shot from the right block to put Ohio State back in front.
Gonzaga came up empty on its next two trips, yet still came within an inch of tying it up again when Kevin Pangos' 3-pointer from the right corner went halfway down before rolling back out. William Buford grabbed the rebound and Sullinger again delivered with a baby hook to push Ohio State's lead to 66-61.
The Bulldogs got no closer. By the time Bell banked in an awkward 3-pointer with 11 seconds left, it was too late.
Gonzaga pushed around West Virginia in a 23-point romp in the second round on Thursday. Sullinger, who watched part of the rout from behind one of the baskets, described it as a "wake-up" call.
The talented sophomore slogged through most of the first 36 minutes, battling foul trouble and Gonzaga's swarming defense. Yet he delivered in the clutch to once again send Gonzaga home before the last weekend of the season.
For all their tournament success, the Bulldogs have yet to make a Final Four. Determined to lead Gonzaga to the one destination that has eluded the program, Sacre gave the Bulldogs an impassioned pep talk in the tunnel before taking the floor just before the opening tip, urging "everybody to hit their man."
Gonzaga landed the first blows, dominating the paint early against the supposedly bigger, deeper Buckeyes.
Ohio State spoke openly about avoiding the sloppy mistakes that plagued the Buckeyes during a lethargic win over Loyola (Md.) in the second round. Sullinger warned his second trip to the tournament would end quickly if Ohio State didn't snap out of it.
The Buckeyes did, but it took a while.
The Bulldogs led by as many as seven in the first half even with Sacre dealing with foul trouble. Sam Dower filled in capably, crashing the glass and even knocking down his third 3-pointer of the season.
Thomas, coming off a career-high 31 points against Loyola, did the same after Sullinger went to the bench with his second foul midway through the half.
Thomas ripped off 10 straight points during a 4-minute stretch as Ohio State clawed back, and Craft erased all of Gonzaga's hard work with a difficult runner just before the horn to give the Buckeyes a 39-37 lead as Ohio State fans who made the 3-hour drive from Columbus roared their approval.