Phoenix won for the sixth time in seven games, overcoming a first-half road deficit to move within two games of the lead in the Pacific Division.
Dragic hit two foul shots to give the Suns a 99-97 lead and his layup with 1:01 remaining boosted the advantage to four.
The only points Denver could produce over the final three minutes came on a dunk by JJ Hickson with 19 seconds to play.
Six Denver players reached double figures in scoring, but none could produce more than the 15 delivered by Wilson Chandler.
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