In the final Big East game between the old rivals, Syracuse bounced back from having lost twice to the Hoyas in the regular season.
Syracuse will take on No. 4 Louisville in Saturday's title game. The Cardinals earned their date with Syracuse by dismissing Notre Dame 69-57.
Fair scored only six points, hitting just 3-of-16 from the field. It was his dunk, however, with 2:04 left in the extra period that gave Syracuse a four-point cushion.
He also stole the ball from Georgetown's Otto Porter with 5 seconds to go to protect Syracuse's three-point lead. Fair, however, then missed two free throws to give the Hoyas a final chance.
Jabril Trawick missed a 3-point attempt at the buzzer that would have forced overtime.
Porter hit two free throws with 7 seconds remaining in the second period to force the overtime.
James Southerland, Brandon Triche and Baye Keita all had 13 points for Syracuse (26-8), which has had to win three games in three days to get to the championship contest.
Georgetown (25-6), swept its two previous games against the Orange by a combined 33 points. The Hoyas were led by 15 points from Mikael Hopkins.
Syracuse and Georgetown played each other in the first Big East Tournament game 33 years ago. Syracuse will leave the league next season for the Atlantic Coast Conference.
Louisville made half its 14 attempts from 3-point range and came up with 20 points from Russ Smith in its win over Notre Dame.
The Fighting Irish hit just 19 field goals for the game.
Louisville (28-5) and Syracuse will be meeting for the third time this season with the teams having split those contests.