Goals from Klose and Candreva ended Roberto Mancini's hopes of steering his team into the competition, and it will have to settle for the Europa league next season.
Juventus, Napoli and Roma have secured Champions League football for the 2016-17 season following Inter's loss to Lazio on Sunday.
Roberto Mancini's side had still been in the hunt for a third-place finish until its most recent defeat, but with only two games left to play it can no longer make up the points required to catch Roma.
Miroslav Klose dealt an early blow to Inter's chances at the Stadio Olimpico when the veteran German striker opened the scoring after just eight minutes.