Kevin Pangos added 15 points for Gonzaga (8-1), which was plagued by poor shooting in the first half and kept 11 points below its scoring average.
Daniel Mullings scored 17 points for New Mexico State (7-4), which lost its third straight game. Tshilidzi Nephawe added 13 points.
Gonzaga leads the nation in field goal shooting at 55.8 percent, but was cold at the start and couldn't put the Aggies away until the final minutes. They shot just 48 percent, but won the rebound battle 45-30. Gonzaga, which led by one point at halftime, shot 64 percent in the decisive second half.
Przemek Karnowski had nine points and nine rebounds for the Zags.
In a seesaw first half, New Mexico State's Sim Bhullar had a tip-in and a stuff as the Aggies built a 10-4 lead in the opening minutes. The Zags made just two of their first 10 shots.
Dower scored six points as Gonzaga came back to tie the score at 16.
Consecutive baskets by K.C. Ross-Miller put the Aggies ahead 24-19.
Gary Bell Jr. and Drew Barham scored to keep Gonzaga close, and a put-back by Coleman gave Gonzaga a 29-28 lead with 3:06 left in the first. It was their first lead since the opening two minutes. Coleman, who had 10 points in the first, added another basket for a 31-28 lead.
New Mexico State was scoreless for more than five minutes before Mullings' basket at the buzzer left the Aggies trailing 31-30 at halftime.
Gonzaga made just 35 percent of its shots in the first half and missed all five of its 3-point attempts. But 14 offensive rebounds led to 13 second-chance points for the Zags. New Mexico State shot 43 percent in the first.
Gonzaga scored the first seven points of the second half, six by Dower, for a 38-30 lead. Pangos hit Gonzaga's first 3-pointer of the game for a 41-33 lead with 16:12 left.
A jumper by Ross-Miller brought the Aggies within four points, but Coleman made three unanswered baskets for a 49-39 Gonzaga lead. While Gonzaga was finding the range, New Mexico State made just three of its first 15 shots in the second.
Pangos' 3-pointer gave Gonzaga a 53-41 lead. The Zags kept a double-digit lead most of the way after that.