UTEP (11-7, 4-1) trailed the entire second half but tied the game five times before Moore hit a jumper to knot it 65-all with 19 seconds showing.
East Carolina (12-7, 3-3) regained the lead, 67-65, on a Miguel Paul layup with 4 seconds left.
UTEP's Jacques Streeter took the ensuing inbounds pass, dished it off to Moore, who managed three dribbles before pulling up midway between the 3-point arc and half-court line for the game-winner.
It was Moore's third trey and the 14th for the Miners, shattering their previous game-high of nine.
UTEP has won 10 of its last 13 games after starting the season 1-4.
Paul led East Carolina with 20 points, but the Pirates couldn't overcome a 9-of-27 first-half shooting performance.
Jalen Rose and David Jacoby announce their choices for sleepers in the 2016 NBA Draft.
Daryl Gross, who spent a decade as Syracuse's athletic director, will start as Division II Cal State-Los Angeles' AD in July.
Utah forward Brekkott Chapman, who averaged 4.4 points and 2.8 rebounds last season, has transferred to Weber State, coach Randy Rahe announced Wednesday.
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