Garnett stopped a 14-3 Hawks run with a jumper, starting a six-point Boston surge that made it 35-27 in the second quarter. And Allen scored 10 straight Celtics points midway through the third that extended a 13-point lead to 67-48 with 5:39 left in the period.
The top-seeded Celtics' lead ranged from 12 to 27 points the rest of the way.
The second game of the best-of-seven series will be in Boston on Wednesday night.
Eighth-seeded Atlanta, which ended the NBA's longest playoff drought with its first appearance in nine years, was led by rookie Al Horford with 20 points and 10 rebounds and Joe Johnson added 19 points.
Garnett had skipped two days of practice for personal reasons before rejoining his teammates Saturday. He didn't miss a beat.
Playing in his first playoff series in four years, the former Minnesota star had 10 rebounds. When he left with 3:26 remaining in the game, fans chanted ``MVP, MVP.''
Paul Pierce scored 16 for Boston and Rajon Rondo had 15.
Celtics center Kendrick Perkins, center, hauls down a rebound between Hawks Josh Smith and Al Horford. Horford lead Hawks scorers with 20. Smith had three blocks. / photo: Associated Press
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