The Orange (23-10) trailed 59-50 with 11:23 to play, but scored 18 straight points to go up 68-59 with 4:59 to play. Daon Merritt broke the run with a free throw at the 4:03 mark. The Jaguars went without a field goal for more than 9 minutes until Ernest Little's dunk with 2:10 left to play.
Nichols, the Big East's leading scorer, gave Syracuse the lead for good on a 3-pointer from the top of the key with 8:05 left.
Syracuse will host a second-round game Monday against San Diego State, which beat Missouri State 74-70 late Wednesday.
Demetric Bennett led South Alabama (20-12) with 23 points, going 7-for-11 from 3-point range. Little finished with 17 points, 18 rebounds and six blocks in the Jaguars' fifth straight loss. Little's 18 rebounds were a Carrier Dome record for an opponent.
Syracuse extended its lead to 72-60 on a pair of free throws by Devendorf with 3:13 remaining.
But South Alabama made it tight in the final 90 seconds, getting a 3-pointer from Bennett and two from Carlos Smith, who cut the Orange lead to 77-73 with his second 3 and finished with 15 points.
South Alabama led 39-38 at halftime on the 35-footer Bennett banked in at the buzzer.
Devendorf, who scored 33 earlier this year against Villanova, scored 18 points in the first half. He also had a team-high six assists.
The Jaguars seemed unfazed by the Orange's 2-3 zone, which limited opponents to 37 percent shooting during the year, third best in the NCAA. South Alabama shot 45 percent, including 6-for-13 from 3-point range in the opening half.
But Syracuse countered South Alabama's run with one of its own, scoring 10 consecutive points to pull to within 22-20 with 8:45 to play in the half.
Nichols hit just one 3-pointer in the first half. He sat the final 5 minutes of the half after picking up his third foul.
Syracuse's defense stiffened in the second half when it forced South Alabama to miss 24 of their 37 shots (35 percent).
It was the first meeting between Syracuse, the second seed in the South Region, and South Alabama, the seventh seed. Syracuse last appeared in the NIT in 2002 and beat St. Bonaventure and Butler at home and Richmond on the road to advance to the semifinals at Madison Square Garden, where the Orange lost to South Carolina.