The defending champion Heat took a 3-0 lead in the Eastern Conference quarterfinal series and can complete a sweep Sunday when the Bucks host Game 4.
Allen made five 3-pointers in Thursday's game, giving him an NBA-record 322 in his 131 career post-season games. He broke the mark owned by Reggie Miller.
Miami trailed by two at the half, but took charge by outscoring the Bucks 30-18 in the third period. The Heat hit 51 percent of their field goal attempts and forced 20 Milwaukee turnovers.
The Heat pulled away even though Dwyane Wade made just 1-of-12 from the field.