The Heat were trailing by one point after Andre Iguodala hit a deep jumper with 14 seconds remaining when LeBron James found Allen open in the left corner.
His shot went through despite being fouled by Corey Brewer and ensuing free throw resulted in a 119-116 lead.
A last-gasp 3-point try by Danilo Gallinari missed and the Heat held on for the victory, which came after a 20-point Friday loss to the New York Knicks.
Chris Bosh pumped in 40 points -- his most with the Heat -- while James contributed 20 points and 11 assists in the win.
The Nuggets, meanwhile, fell to 0-3 despite 22 points each from Iguodala and Kenneth Faried.