His early tennis experiences were at the shell-ravaged Zetra Olympic Hall which was used as a temporary morgue during the war.
On Friday, the 23-year-old Bosnian will face his boyhood idol Roger Federer, probably on Court Phillipe Chatrier, in the French Open third round.
"Probably the biggest player ever in tennis, and I'm really glad I have the chance to play with him," world number 88 Dzumhur told reporters after beating Cypriot Marcos Baghdatis on Wednesday.
"That is probably the dream of every player and I have this opportunity now.
"I remember when I played a tournament -- was it under-12 or under-10? -- I watched him playing semis, I think Wimbledon, with (Andy) Roddick. That was the first time I was cheering for him. Since then he's my idol."
Dzumhur became the first player to represent Bosnia in the main draw when he qualified for last year's Australian Open and reached the third round.
It was a remarkable effort considering the shoestring budget of Bosnia's tennis federation and lack of facilities in a country recovering from the bitter 1990s conflict.
"I'm really happy I can represent my country and the name and the flag of my country is on these big tournaments," he said.
"I have some friends of course they're playing. I'm sure that we have a lot of talents, but unfortunately in Bosnia, nobody cares about the sport actually."
Still living in Sarajevo, he is coached by his father Nerfid and has the same tennis racket sponsorship with Wilson, who also supply Federer.
Apart from that, nothing has been given -- a far cry from players from some of the richer tennis federations.
"You have to go through everything alone," he said. "I say alone, but with my parents, of course."
At home he has to join the queue for practice courts, especially in the winter months.
"All the clubs have to use the same courts, it's always like that.
"In April I had to play clay tournaments and there was still balloons over most of the courts in Sarajevo. I had to find any court that I could play outdoor. We have only one hard court that we can really play."
Image: Damir Dzumhur
Photograph: Mike Ehrmann/Getty Images