Both sides struggled to create chances in the first half, although the hosts could have taken a shock early lead as Amadou Diawara played Anthony Mounier in at the back post, but the French winger was unable to connect with the pass.
Inter's best chances came from free kicks, as Ivan Perisic headed an Adem Ljajic delivery wide, before the Serbian forward struck a dipping effort from 30 yards straight at Bologna keeper Angelo da Costa.
After the break Ljajic came close to breaking the deadlock after former Newcastle United full-back Davide Santon found the forward with an inviting cross, but the 24-year-old's effort was blocked.
The visitors took the lead on 67 minutes, as Ljajic found space in the Bologna box following a defensive mix-up before finding striker Mauro Icardi for a tap-in from close range.
Despite being a man down Inter improved after finding a goal, and almost doubled their advantage as Icardi dragged an effort wide in the path of Freddy Guarin, who saw a curling effort deflect behind for a corner.