Trailing by seven with just over 5 minutes left, Syracuse tied it at 55 on Michael Carter-Williams' 3-pointer from the top of the key with 80 seconds left.
Fair's tip came after Jerami Grant drove the lane and missed. Cincinnati's Cashmere Wright missed a 3 from straight on with 2.9 seconds left and the Orange had their 35th straight win at home, the longest active streak in Division I.
The Orange outscored the Bearcats 13-4 in the final minutes. Fair finished with 13 points.
Syracuse (18-1, 6-0 Big East) was coming off a 70-68 win at Louisville on Saturday that knocked the Cardinals out of the No. 1 spot in the nation. The Orange, tied for third in the rankings with Kansas, are the only Big East team with an unblemished conference record.
Syracuse is 28-1 in regular-season play in the Big East in the past two years, that lone loss coming at Notre Dame exactly one year ago Monday.
The Bearcats (16-4, 4-3 Big East) had won three in a row. They lost on the road for the first time in eight games.
Carter-Williams had 16 points and seven assists and Brandon Triche had 13 points for the Orange.
Wright, who sprained his right knee Tuesday in a win over DePaul and did not play in Cincinnati's overtime victory against Marquette on Saturday, finished with five points on 2-for-13 shooting. He was 1 of 8 on 3s.
Sean Kilpatrick led Cincinnati with 21 points and JaQuon Parker had 11.
Syracuse's James Southerland, tied for second on the team in scoring at 13.6 points per game, missed his third straight game because of an eligibility issue. He was in street clothes on the bench.
Cincinnati played the shot-clock game with its deliberate half-court approach and kept the game close despite a poor shooting performance in the opening half.
Trailing by just a basket at the break, the Bearcats started the second with a 12-2 spurt keyed by consecutive 3-pointers from Parker when he was left unguarded in the right corner.
A high-arcing 3 by Kilpatrick was the fourth make in five tries from long range for the Bearcats and gave them a 36-29 lead with 13:46 left.
Carter-Williams responded with seven straight points, his 3 from the top of the key making it 36-all midway through the half.
Undaunted, the Bearcats kept charging, and Kilpatrick's sixth 3-pointer of the game and a layup by Cheikh Mbodj completed an 11-4 spurt that gave them a 49-42 lead with 5:44 left.
Triche, whose helped boost Syracuse at Louisville, then hit two straight jumpers and fed Grant for a slam dunk to draw the Orange within 54-50 with 3:17 left.