Craft missed from almost the same spot with 31.4 seconds remaining, but the scramble for the rebound went out of bounds off the Cyclones, giving the Buckeyes one final possession with 29.9 seconds left and the game tied at 75.
The fact that the second-seeded Buckeyes needed last-second heroics from Craft was because they squandered a 69-56 lead with 6:04 remaining. But Iowa State, the West region's No. 10 seed, went on a 10-0 run to tie the game at the 3:53 mark.
From there, it was a classic college basketball game with each team trying for the knockout punch until Craft delivered the late haymaker.
Junior swingman Deshaun Thomas led all scorers for the Buckeyes with 22 points while Craft added 18 to go with six assists and two steals. Sophomore guard LaQuinton Ross also hit double figures with 17 points, 10 of them coming in the second half.
Iowa State had four players in double figures, led by senior guard Korie Lucious, who connected on 5 of 7 treys and finished with 19 points. Senior guard Will Clyburn tallied 17 while backup guard Tyrus McGee came off the bench to score 14.
Junior forward Melvin Ejim added 10 points and a game-high 11 rebounds, but the Cyclones failed to get much production from freshman Georges Niang. The 6-7 forward tied his career-high of 19 points during Iowa State's 76-58 win over Notre Dame in the team's tournament opener, but Niang managed only eight points of 3-of-6 shooting against the Buckeyes. Niang was 9 for 13 against the Fighting Irish.
The Cyclones, the nation's leader in three-pointers per game and total treys made this season, stayed in the game because of their outside shooting. They fired up 25 three-pointers in the game and made 12 for a sizzling 48.0 percent behind the arc.
But Iowa State was done in by its inability to make shots closer to the basket. ISU missed at least a half-dozen layups in the contest and finished 23 for 52 (44.2 percent) overall.
Meanwhile, the Buckeyes connected on 25 of 49 shots (51.0 percent), including a 12-for-19 performance (63.2 percent) in the second half. OSU also connected on 50.0 percent of its three-point shots (9 for 18).
Neither team was able to gain much separation during a fast-paced first half that featured 10 lead changes and seven ties.
Ohio State drew first blood when Thomas scored just 23 seconds into the game, but Iowa State responded with nine of the 11 points to take a 9-4 edge at the 15:23 mark.
The Buckeyes quickly erased that deficit with a 10-0 run over the next three minutes, a spurt that featured a Craft flash to the basket for a layup and a Scott steal that resulted in another layup. Unfortunately, Scott ended his own team's momentum by drawing a technical foul for taunting ISU defender Bubu Palo.
Babb connected on both technical foul shots and followed with a trey to tie the game at 14 with 11:37 remaining to the break.
OSU regained control of the game with a 9-4 run over the next three minutes to take a 23-18 lead at the 8:49 mark, but the Cyclones finally got hot from three-point range and the first half remained a pitched battle the rest of the way with neither team gaining more than a two-point advantage during the final 6½ minutes.
The Buckeyes fired the final salvos of the first half when Thomas slashed to the rim for a layup at the 1:08 mark to tie it at 36 and then Ross put on a shake-and-bake move for another lay-in to give Ohio State a 38-36 lead at intermission.
Things stayed tight in the early going of the second half until the Buckeyes got a little breathing room after junior guard Lenzelle Smith Jr. hit a three ball from the right corner to give his team a 52-46 advantage at the 14:12 mark.
The Cyclones fought back to within one at 52-51 when Ejim flushed a two-handed dunk off an offensive rebound, eliciting a timeout from OSU head coach Thad Matta with 11:43 to play.
Whatever Matta said during the break rallied the Buckeyes, who scored 17 of the next 22 points to grab what looked like a comfortable 69-56 lead with 6:04 remaining.
But back came Iowa State with a 10-0 run to tie things at 69 with 3:53 showing on the clock.
Thomas connected on a pair of free throws to give the Buckeyes the lead back at 71-69, but McGee fired in a three at the 3:24 mark to put the Cyclones back ahead at 72-71. McGee hit one of two free throws 24 seconds later to push his team's lead to two, but Craft slashed to the basket and got an old-fashioned three-point play to put Ohio State back on top at 74-72 with 2:35 to play.
Iowa State got its final lead of the game when McGee bagged two more free throws with 2:22 to go, but the Cyclones missed a golden opportunity to add to their 75-74 edge when Clyburn was called for charging at the 1:41 mark, erasing a two-pointer.
Back the other way Ohio State came with Craft drawing another foul at the 1:05 mark. But he made only one of his two attempts at the line, tying the game for the 16th and final time.
Ejim turned the ball over on Iowa State's next possession and the Cyclones never got it back.
Craft missed his three-point try with 31.4 seconds, but he got another chance after the rebound went off an Iowa State player. The OSU point guard dribbled away nearly all of the 29.9 seconds remaining before pulling up just in front of the arc at the top of the key and sending the Buckeyes to the Sweet 16.