At least two civilians and five attackers died in the assaults, described as an attempt to mimic the deadly Paris attacks.
Security forces battled militants for hours in a major business and shopping district.
It follows warnings late last year that Islamists were planning a major attack.
IS released a statement online claiming the attacks, which it said were carried out by "soldiers of the Caliphate", targeting "citizens of the Crusader coalition" against the group.
