Marvin Bagley III scored 22 points and Wendell Carter Jr had 14 with 12 rebounds for the Blue Devils (29-7), who will face top-seeded Kansas - a rare matchup of bluebloods in a tournament defined by chaos - on Sunday in Omaha.
But Duke couldn't shake the Orange (23-14) until the final buzzer sounded.
"It was a hard-fought game," Bagley said.
Sharp-shooting Grayson Allen missed the front end of a one-and-one with 12 seconds left and Duke up 67-64.
The Blue Devils fouled rather than let Syracuse go for a game-tying 3, and Tyus Battle only made one of two from the line.
Gary Trent Jr. then made two free throws with 6.3 seconds to go to seal a closer-than-expected win that was similar to the Jayhawks' 80-76 win over Clemson - a game Kansas led by 20 at one point.
"We've got to come out ready from the beginning," Bagley said of Kansas. "I think we'll come out ready."
Battle had 19 points to lead the Orange. Allen had 15 for Duke, but he was just 3 of 14 on 3s.
Duke held a 50-45 lead midway through the second half, despite allowing a pair of quick baskets that so infuriated Coach Mike Krzyzewski that he ripped his jacket off in frustration - drawing oohs and aahs from a crowd full of blue-clad Blue Devils fans.
The win gave Krzyzewski his 1,100th career victory.
In the NCAA East Region, No. 1 seed Villanova's 3-point party rolled past the intense pressure of West Virginia, 90-78, to bring the Wildcats to the doorstep of another Final Four two seasons after winning a national championship. No. 3 seed Texas Tech eliminated No. 2 seed Purdue 78-65.