Boston knocked the Hawks out of the post-season in six games and set up an Eastern Conference semifinal series against Philadelphia, which eliminated top-seeded Chicago earlier Thursday.
Philadelphia defeated Boston in two of their three regular-season meetings.
Atlanta took a 79-76 lead on Al Horford's layup with 2:27 to play and then scored only one point the rest of the way. Garnett's turnaround jumper put Boston in front 80-79 and that lead was expanded when Ray Allen made 1-of-2 foul shots with 9 seconds remaining.
Horford went to the foul line with 2 seconds remaining with a chance to tie the game, but he missed the first free throw.
Rather than miss the second attempt on purpose and hope for a rebound, Horford made good on the other attempt. The Hawks then had to foul Paul Pierce immediately after the inbounds pass and Pierce hit both shots to account for the final score.
Garnett scored 28 points and had 14 rebounds. He was the only Boston starter to make more than half his shots from the field.
Both teams shot just 41 percent.
Josh Smith led Atlanta's balanced attack with 18 points while Joe Johnson had 17.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys