Bosnia were held 0-0 by Portugal in the first leg of their winner-takes-all play-off and now must travel to Lisbon in the return game next Tuesday.
Dzeko had a quiet evening and wasted a late opportunity to give the Bosnians a precious lead to take into the second leg, where the Portuguese must now start as strong favourites.
Savic’s Montenegro have a mountain to climb if they are to progress. They lost 2-0 to the Czech Republic at the Generali Arena, conceding a gut-wrenching second in injury time after putting up a sterling defensive display.
Savic, who plays at left-back for his country, had a solid 90 minutes but with a 3-0 win needed in Montenegro, the tiny Balkan nation’s adventure would appear to be over.
The second legs will be played next Tuesday - former City stars Richard Dunne and Shay Given fared better as Ireland beat Estonia 4-0 away and Croatia won their first leg against Turkey 3-0.