Boston led 89-80 with 5:44 to play before Elton Brand threw down a dunk to spark a 10-0 Atlanta run, which Korver capped with a triple to give the Hawks a 90-89 edge.
Jared Sullinger responded with a trey at the other end to put Boston back in front, but Korver countered with another 3-pointer, igniting a 9-0 burst that provided the home team with a 99-92 lead with 1:13 remaining.
Atlanta sealed the outcome by going 6-for-6 from the free-throw line down the stretch, as the Hawks extended their lead to two games over the idle New York Knicks for the eighth spot in the Eastern Conference standings.
Jeff Teague totaled 19 points and eight assists, while Paul Millsap and Brand each donated a double-double. Millsap supplied 14 points and 14 rebounds, while Brand netted 13 points and grabbed 13 boards in Atlanta's third win in four tries.
Avery Bradley returned from a three-game absence and paced Boston with 24 points, while Rajon Rondo registered 19 points and 12 assists for the Celtics, who have lost 13 straight road games, their most in a single season since 1979.