Hawkins, who finished with a double-double of 16 points and 13 rebounds,was fouled by UTEP's Stefon Jackson on an inbounds play after the Miners turned the ball over while trying to get past the Aggies' fullcourt press.
After Hawkins hit the two free throws, UTEP guard Marvin Kilgore missed on a 22-foot 3-pointer.
Jackson led UTEP with 16 points and Malik Alvin scored 12. Maurice Thomas had 11 rebounds and Victor Ramalho grabbed 10 rebounds.
David Fisher led the Aggies with 18 points.
There were 15 lead changes in the second half. The Miners took a 68-66 lead with 1:16 left on a 3-point play by Ramalho, who as fouled by Hatila Passos.
Fred Peete hit one of two free throws for NMSU with 1:08 remaining and UTEP's Pearson Smith hit the second of two free throws with 22.4 seconds remaining for a 69-67 UTEP lead. Peete's two free throws with 7.7 second left tied it.
UTEP got the ball in but were the Miners were called for a turnover. Hawkins was fouled as the Aggies tried to inbound their possession.
UTEP took a 31-22 lead with a 13-0 run in which Alvin scored 7 points with five free throws and a goal tending basket.
New Mexico State closed the half with an 8-1 run to get within 32-30. Neither team led by more than four points in the second half.
In a game where play was frequently stopped by fouls and sloppy play, the Miners had 27 turnovers and New Mexico State 21.