LeBron James finished with 32 for the Heat.
Miami started quickly.
The Heat converted seven of their first 10 possessions and, behind five points by James, built an early 16-8 lead.
Utah stayed close on back-to-back baskets by Paul Millsap, however, and DeMarre Carroll gave the Jazz a huge lift off the bench.
Carroll scored seven points in a two-minute span and Utah opened a 30-25 lead at the end of the first quarter.
The Jazz kept rolling and start the second quarter, when they began with a 15-4 run. Carroll scored five points and Hayward buried a three-pointer as Utah built a 45-29 lead.
The Jazz stayed in control and took a 59-44 halftime lead, but they actually helped Miami stay in the game by committing six turnovers in the final four minutes.
James scored 20 in the first half on 8-for-11 shooting. No other Heat player had more than eight.
Hayward and Carroll topped the Jazz with 12 points each.
In the third quarter, Utah owned an 65-50 lead before converting nine of 10 possessions.
The Jazz opened an 80-60, but the miss five of their final six shots and the Heat stay alive heading into the final 12 minutes.