Jayson Tatum scored 31 points and pulled down 10 rebounds and Jaylen Brown added 28 points as six Celtics players scored in double figures and Boston cut the deficit in the best-of-seven Eastern Conference finals to three games to two.
With their season hanging in the balance, the Celtics exploded for 41 points in the third quarter to turn the tide in a game dominated early by Miami.
Under solid pressure from the Heat defense, the Celtics made just one of their first 12 shots and connected on just five of 20 from the field in the first quarter.
Boston trailed by as many as 12 in the opening period, which ended with the Heat leading 26-18.
Miami again built the lead to 12 in the second period and finished the first half with a 58-51 lead.
But Boston were beginning to see some shots fall, and as they turned up the intensity on both ends of the floor after the interval Miami had no answer.
They wouldn't trail again.
Walker's three-pointer with 4:26 left in the third gave Boston a 10-point lead and the Celtics took a nine-point edge into the final frame, when they continued to pour it on.
The Celtics will try to build on their momentum against a Heat team determined to put them away in game six on Sunday.
The Lakers lead the Nuggets three games to one in the Western Conference finals.