Devin Davis had a chance to seal the win, but the Cougars' gritty forward missed a pair of foul shots with 3.6 seconds left. The Wolverines (30-7) called timeout to set up a final play, and Muhammad-Ali Abdur-Rahkman found Poole on the wing, and the shot hit nothing but net.
The officials reviewed it to make sure, but Poole had clearly gotten the shot away.
Abdur-Rahkman and Moe Wagner scored 12 points apiece for Michigan, but it was the unheralded freshman who stole the show. Poole's flair for the dramatic earned coach John Beilein's team a trip to Los Angeles for a West Regional semifinal against North Carolina or Texas A&M next week.
Rob Gray scored 23 points and Davis finished with 17 for the Cougars (27-8), who were trying to reach their first Sweet 16 since the last of the Phi Slama Jama teams went to the Final Four in 1984.
They just about did it.
Davis gave the Cougars the lead when he made two free throws with 44.1 seconds left. He pushed the advantage to 63-61 when he made the second of two more foul shots with 24.9 seconds to go.
Abdur-Rahkman and Charles Matthews came up empty at the other end for Michigan, and Davis pulled down a crucial rebound, and then stalked to the foul line.
The senior forward missed both.
The down-to-the-wire outcome was hardly surprising given the way the rest of the game went. There were 17 lead changes and 12 ties, including 28-all at halftime.
After his huge performance against San Diego State, the Wolverines were wary of Gray every time he touched the ball. They blanketed Houston's star on the perimeter, cut off lanes to the basket and held him to just eight points on 2-for-11 shooting in the first half.
Wagner continued to struggle after a poor NCAA Tournament opener, scoring just three points while dealing with foul trouble. Abdur-Rahkman was 2 of 8 from the field, including 0 of 5 from the arc.
Michigan air-balled three 3-pointers in four possessions spanning the break.
Whistles became constant as the second half wore on, and both teams soon found themselves in foul trouble. Wagner picked up his fourth with 8:43 to go, and Breaon Brady soon took a seat with his fourth for Houston as the game turned into a glorified free-throw shooting contest.
Michigan converted eight straight at one point to take a 57-53 lead with 3:42 to go.
Armoni Brooks answered with a 3-pointer, and Davis converted a three-point play after fouling Duncan Robinson out with 2:06 left, giving the Cougars a 60-59 lead.
Wagner answered with a putback basket for Michigan with 1:41 left, but after the teams swapped 3-point misses, Davis grabbed a crucial rebound and made two foul shots to give Houston the lead.
His night would have been a whole lot better if the game ended there.
Houston showed remarkable poise down the stretch, led by a pair of seniors and three juniors in its starting lineup. But the cracks showed when Davis went to the foul line with a chance to ice it.
Michigan won its 11th straight game in the most dramatic of fashions, even without its top players at their best. Abdur-Rahkman finished 4 of 15 from the field and Matthews was 5 for 12.
Michigan heads to the West Region semifinals in Los Angeles, where it will face the winner of Sunday's game between the second-seeded Tar Heels and No. 7 seed Aggies.
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