The Lobos (8-4, 5-2 Mountain West) shutout the Cowboys (4-7, 2-4) over the final 56 minutes.
New Mexico piled up 413 yards, including 336 on the ground, and held Wyoming to 122 yards - only 23 rushing.
It was 9 degrees with a wind chill of minus-5 when the game kicked off, and by the end of the contest the wind chill was minus-14. Snow fell throughout the day.
Wyoming scored on its opening drive, a 22-yard field goal by Deric Yaussi, but New Mexico answered with an 11-play, 82-yard drive capped by Kelly's 9-yard TD pass to Moore with 6:46 left in the opening period.
On Wyoming's next possession, Lobos lineman D.J. Renteria sacked Cowboys quarterback Casey Bramlet, forcing a fumble recovered by New Mexico's Sidney Wiley at the Wyoming 25. That led to Wes Zunker's 24-yard field goal on the first play of the second quarter to make it 10-3.
Kelly's 12-yard scoring pass to Byrd with 6:54 left in the third put the Lobos up 17-3.
In the final period, New Mexico picked up a safety when Bramlet was whistled for intentional grounding from his own end zone after being harassed by Nick Speegle. The Lobos closed scoring with a 1-yard run by Cox with 11:36 remaining.