The Celtics led 73-58 at the end of the run. Joe Johnson, who had 25 points, had three late 3-pointers as the Hawks rallied. Johnson’s 3 with 11.8 seconds remaining cut the lead to 77-76.
Allen, who had 19 points, made two free throws with 10.9 seconds remaining.
Jeff Teague launched an air ball on Atlanta’s final possession.
Kevin Garnett had 16 points and Rajon Rondo had 10 points, 13 assists and six steals for Boston, which took charge with a 17-2 run, including 13 unanswered points, after a 56-56 tie in the final period.
Zaza Pachulia had 16 points and 13 rebounds for Atlanta, which now leads Boston by 1 games in the race for the sixth spot in the Eastern Conference.
Paul Pierce had 13 points for Boston.
The Celtics managed only 31 points in the first half and shot only 35 percent from the field through three quarters to trail the Hawks 51-50. The Celtics broke through with 29 points in the final period.
Each team was short-handed. The Celtics announced before the game that forward-center Jermaine O’Neal will have surgery on his left wrist and miss the remainder of the season, adding more urgency to the team’s search for inside help.
Hawks guard Jannero Pargo had an appendectomy Sunday night in Cleveland and is expected to miss two to four weeks.
The Hawks also were without forwards Marvin Williams (hip) and Vladimir Radmanovich (back) and guard Willie Green (hamstring).
Each team struggled with poor shooting in the first half. The Hawks matched with season low with 15 first-quarter points and set a low for points in a first half but led the Celtics 33-31.
Fouls became a growing problem for Boston in the third period. Pierce and Brandon Bass drew their fourth fouls with a span of 2 minutes midway through the period.
Bass drew a flagrant one technical foul for knocking Pachulia to the floor with 7 minutes remaining in the third. Less than a minute later, Pachulia again was knocked to the floor again by an elbow thrown by Greg Stiemsma under the basket. No foul was called, but Pachulia was bleeding badly from a cut on the top of his head, and he had to be taken to the locker room.
Stiemsma remained in after he was called for his fourth foul with 3 minutes remaining in the period.
Pachulia, who needed stitches, returned for the start of the fourth period.