Haynes hit a trey from the wing and was fouled on the play by Malerick Bedden, then hit the free throw.
NMSU is now 15-6 overall and 6-4 in the Sun Belt, good for second in the Western Division, while South Alabama falls to 11-11 and 5-6.
The Aggies were down 71-69 after USA's Henry Williams scored to put the Jags up with 1:21 left.
NMSU called timeout and actually drew a play for forward James Moore. But USA wasn't about to let Moore, who scored 23, beat them as it played tight post defense.
Instead for the second time in the last couple of weeks, Haynes hit a big triple. Haynes also hit a big trey in a win at Denver last Thursday to give NMSU a late lead.
"The play was set up for James, but I told the guys I'd be ready if the ball came to me," Haynes said. "I got a good look and hit the shot. And on the free throw, I knew that was a big one. I concentrated and I'm glad I hit because I had been struggling from the line."
USA had chances to tie or go ahead down the stretch, but Bedden's runner was off with 31 seconds left. Jason Fontenet put NMSU up four, 75-71, with 27.4 seconds with two free throws.
Then the Jags missed three 3-pointers and Moore added two final free throws to close out the game.
"I thought we played pretty good defense down the stretch," NMSU coach Lou Henson said. "We didn't give them good looks and we hit our free throws."
The Aggies hit 21 of 22 from the charity stripe for 95.5 percent, setting a new school record for best percentage.
NMSU also got solid showings from Moore and senior wing Brandon Mason on the night.
Moore led all scorers with 23 points as he hit 7 of 11 from the field and 9 of 9 from the charity stripe. Mason hit for 22, including 9 of 10 from the field.
Mason was 35 of 92 in conference play before Saturday's game, while Moore had struggled in the past two games, making just 5 of 17 from the field.
"I was more aggressive," Moore said. "I got some easy baskets and got to the free throw line."
Mason's 22, included a huge 3-pointer with 1:37 left that knotted the game at 69 as the shot clock was winding down.
"I felt good today, I was relaxed," Mason said. "On that late 3-pointer, the clock was winding down and I just felt good when I shot it."
NMSU had to overcome the sharpshooting Jaguars, who hit 10 of 20 from beyond the 3-point arc. At one point in the game, USA had hit 10 of 13.
Junior guard Chris Young, who's been marred in a slump as of late, came out of it in a big way, hitting 5 of 10 3-pointers for 19 points. Forward Richard Law added three triples and had 11.
"They shot the ball really well," said Henson of the Jaguars. "We played well tonight and we had to."
NMSU shot 52 percent and out-rebounded the Jags 31-25 in a game that featured 14 lead changes and seven ties.
The first half featured timely shooting by both teams. USA made 7 of 10 treys, three by Young as it took a 37-35 lead into the break. He had 11 in the opening half.
NMSU made 15 of 23 from the field for 65.2 percent and was led by Mason's 11 and Moore's 10. Mason was 5 of 5 in the first half.
In the second half, both teams cooled just a bit, as the lead see-sawed back and forth.
The largest lead of the second half was the final margin of six.
Henry Williams paced USA in the second half with 17 of his 19 as he continually worked the paint with solid post moves and offensive rebound. Williams had five offensive rebounds in the second half, including a big stretch in the final seven minutes of the game.
He scored 12 points in that stretch.
But the Ags countered with Moore, Mason and Jason Fontenet. Moore hit for 13, Fontenet had all 12 of his points and Mason had 11.
The Aggies also hit 16 of 16 from the free throw line in the second half.
"We got a good effort out of James and Brandon," Henson said. "We did a good job of shooting the ball and working our offense."
Said USA coach John Pelphrey: "We needed a couple of stops, but you have to give NMSU credit down the stretch. I'm proud of our effort on this trip. We played well at (Louisiana) but needed more execution on offense there."
NMSU is back in action Monday in non-conference play against Colorado-Colorado Springs.
Game time is set for 7:05 p.m.