Deshawn Delaney added 15 points for the Lobos (11-5, 3-1 Mountain West Conference).
Greenwood's final 3-pointer came as the shot clock sounded with 32 seconds to play, extending the Lobos' lead to 65-60 and frustrating the Aggies' comeback efforts. Greenwood also hit two technical free throws with 1:35 to play to break a 60-all tie.
USU briefly took the lead at 58-57 on Moore's three-point play with 2:51 to play, their only lead of a tight second half.
New Mexico outrebounded Utah State 36-22, including 14 offensive rebounds. The Lobos dominated scoring in the paint 32-16 and had 17 second-chance points.
USU only shot 55 percent from the free-throw line, including 2 of 7 in the first half. USU came into the game as the poorest free-throw shooting team in the conference at 45 percent in conference games.
Utah State led much of the first half, but Greenwood nailed three 3-point shots and a pair of free throws in the final minutes of the period to give the Lobos a 34-31 halftime lead.