In a result that extends Atalanta's winning run to four Serie A games, Bonaventura scored twice, with his winning goal coming after withstanding a second-half barrage from Inter.
However, Inter's profligacy and Bonaventura's brace proved crucial as the hosts missed the chance to move ahead of fourth-placed Fiorentina, who make the trip to Napoli later on Sunday.
Inter, though, failed to make the most of their chances and Bonaventura made them pay - heading home in the 90th minute to extend Inter's winless run against Atalanta to six matches.
With Walter Mazzarri's Inter going in search of a third straight victory, Hernanes played in Fredy Guarin for the game's first opportunity - the Colombian placing an effort marginally wide of Andrea Consigli's left-hand post.
Guarin's third effort stung the palms of Consigli before Icardi failed to provide a cutback for Rodrigo Palacio when played in behind the Atalanta defence.
Bonaventura's long-range strike dipped too late to threaten Handanovic before he eventually found the net at the third attempt after 35 minutes.
Following a quick break, Bonaventura made the most of Denis' decoy run to create space before firing past Handanovic.
However, Atalanta's joy was shortlived, as a minute later Icardi collected Guarin's slipped pass before cutting inside his marker and firing home the equaliser to restore parity nine minutes before the break.
As in the first half, Inter began on the front foot after the interval with Guarin at the centre of their attacking threat - firing a shot from distance that clipped the crossbar and went over.
Palacio was then denied by the post as he glanced Jonathan's cross over the helpless Consigli as Inter continued to pile on pressure.
However, Stefano Colantuono's charges secured a dramatic late win thanks to Bonaventura's fifth league goal of the season as Inter's European aspirations were dealt a blow.
