The Bosnian double-winners are managed by former Hearts striker Husref Musemic and will be Celtic's first opponents in the 2019/20 season.
The first tie will take place at Celtic Park on either 9 or 10 July. The second leg will then take place the following midweek.
Sarajevo were one of four potential ties for Celtic going into the draw. The others were Valleta of Malta, Montenegrin champions Sutjeska, Partizani of Albania or Georgian side Saburtalo.
Celtic will learn their potential second-round opponents tomorrow afternoon.
The Ladbrokes Premiership champions, and winners of a 'treble treble', have already returned for pre-season training in preparation for the clash with Sarajevo. They'll be looking to make the group stage of the Champions League after missing out last season.