Pascal Siakam contributed 14 points for New Mexico State (8-9).
The Aggies, who have lost all seven road games, have also struggled at home this year, losing two for the first time since the 2011-12 season.
The Cowboys' Kendrick Nichols hit a 3-pointer for a 46-42 lead early in the second half. Then NMSU went on a 14-5 run, which included a Siakam jumper, 3-pointers by Eldridge and Ian Baker, a Siakam layup, and four straight points by Barry, to give the Aggies a 56-51 lead at the 11:08 mark. The Cowboys battled back but could never regain the lead.
Nichols had 24 points and Ricky Norris chipped in 17 for New Mexico Highlands.