Brown poured in 26 points as he went 11-for-17 shooting, but it was his aggression and mammoth defensive effort that really impressed as he stole three times and blocked once.
He was ably assisted by Jayson Tatum's 25 points and 14 rebounds, while Kemba Walker contributed 21 points.
Boston needed the win, which took the series to 2-1 in Miami's favour, after stumbling in their previous two outings when the Heat mounted impressive comebacks.
"Our backs were against the wall. We were down 0-2, and I think we should play like that all the time, like we're fighting for our lives," Tatum told reporters after the match.
The Heat will rue their inaccuracy after shooting just 33-from-85, though centre Bam Adebayo had another good day with 27 points and 16 rebounds.