Henderson had 18 of his 19 points after halftime for the Blue Devils (20-3, 7-2 Atlantic Coast Conference), who struggled the entire way before controlling the extra period to bounce back from a blowout loss at Clemson.
Duke trailed by 13 at halftime and 38-22 early in the second half before rallying.
Jack McClinton scored 34 points to lead Miami (15-8, 4-6), including a 3-pointer to force overtime with 14.8 seconds left. Duke had a chance for the win in regulation, but Scheyer missed a 3 in the final seconds.
Once in overtime, Henderson scored over Dwayne Collins inside for a 70-68 lead, then Singler — who finished 5-for-23 from the field — drove in for a score two possessions later to push the margin to four. Then, Henderson added a tough stepback jumper over Collins to make it 74-68 with 51.9 seconds to play.
The Hurricanes, meanwhile, missed their first five shots before McClinton buried a 3 with 31.1 seconds left that cut the lead to three. Miami got no closer, with Greg Paulus burying two free throws with 10.2 seconds left to seal the win.
Paulus, making his first start at the point in nearly two months, finished with 18 points, while Singler had 17. Duke also got a boost from reserve David McClure, who had 13 rebounds — including six on the offensive glass after halftime.
The Blue Devils were coming off a 74-47 loss at No. 10 Clemson on Wednesday night, their worst defeat since a 30-point loss to UNLV in the 1990 NCAA championship game. And considering their next game is at Cameron Indoor Stadium against No. 3 North Carolina on Wednesday night, Duke needed this win to find its rhythm again.
Duke won despite shooting 33 percent from the floor — 19 percent in the first half — and going 12-for-39 from 3-point range against the Hurricanes' determined zone defense.
The Hurricanes were coming off a notable performance of their own, though theirs was worth remembering. Miami turned a three-point halftime lead against seventh-ranked Wake Forest into a 79-52 win to end a three-game losing streak.
They nearly followed up with another impressive victory, but ended up with their third overtime loss in the past five games.