The film had already won the best foreign film award at the Golden Globes, making it one of the favourites for the Oscar.
It unseated the other reigning favourite, the French movie Amelie.
No Man's Land is a bitter-sweet anti-war film about two soldiers, one Bosnian Muslim, the other Serb, who end up in an abandoned trench between the frontlines during the country's 1992-95 war.
The movie, which is written and directed by Tanovic, has been a massive hit in Bosnia, and an art house success worldwide.
Benjamin Filipovic, who is head of the Bosnian Cinematographers' Association, said the world's recognition of the film is a great coup.
He said: "Bosnian cinematography is now recognized thanks to Danis Tanovic. It helps us access foreign producers with our ideas.
"It is like getting into a slipstream of the leader in Formula One races."
Tanovic, who now lives in France, made the film with French and Italian money.
The director was born in 1969 in Zenica in Bosnia-Herzegovina and studied at the Theatre Academy in Sarajevo from 1989 to 1993.
During the first two years of the Bosnian war, Tanovic was responsible for the army archive of Bosnia-Herzegovina and made 300 hours of documentary material.
'Universal message'
He has previously received awards for his films and documentaries in Dublin, Ireland, Fribourg, Germany and in Auxerre and Paris in France.
His father Mehmed is delighted with his son's success.
He said: "When he was leaving Sarajevo in 1994 he said he had all he needed with him. Now he is proven to have been right.
"All he took was his knowledge and it helped him make a film with a universal message."
No Man's Land faced competition from Amelie, Elling from Norway, Lagaan, representing India, and Argentina's Son of the Bride.