Italy won their semi-final against Northern Ireland on Thursday, and a number of players - including Dzeko's former Inter Milan teammate Federico Dimarco - were shown on Italian TV watching Bosnia's penalty shootout win.
The Italians appeared jubilant after Bosnia prevailed, and Bosnian fans reacted angrily when the video was shared on social media, saying it showed disrespect towards their team.
"What happened is something completely normal because we all have our preferences, mine was maybe not to play against Italy," Dzeko told reporters on Monday ahead of Tuesday's game in Zenica.
"But today you have to be careful, with social media everything becomes bigger than it is. Dimarco messaged me and said he didn't want to offend anyone. I told him 'what are we even talking about? There's absolutely no problem'."
Italy risk missing out on World Cup qualification for a third consecutive time.
"Italy didn't want to play in Wales, I don't know why, because we went there without fear and won," Dzeko said.
"I don't know why Italy should be afraid of Wales or Bosnia. Italy is an incredible national team that has won four World Cups.
"If they're afraid of playing in Wales, it means something isn't working, and maybe we should look at it this way, they might really suffer in this match because they're playing for so much after missing two World Cups. That means they're afraid."
Italy's woes do not just concern the national team. No Italian club reached the last eight of this season's Champions League, and having played for Inter, AS Roma and Fiorentina, Dzeko is familiar with Italian football's problems.
Bosnia, meanwhile, are also aiming to return to the World Cup for the first time since 2014, their one and only appearance to date.
"Their media complain all the time that Italian football lacks intensity, that's why Italian clubs haven't been getting good results in Europe lately," Dzeko said.
"I expect a very tactical match, given what's at stake for both and I believe both teams will be very cautious from the start."
Still scoring after all these years
Dzeko netted the equaliser against Wales, maintaining his record of scoring in every year since making his Bosnia debut in 2007. The 40-year-old, his country's all-time top scorer and most capped player, has no intention of stopping just yet.
"I didn't even realise I had scored those goals for 20 years in a row, it's a sign that a player's career flies by in a second," Dzeko, currently playing with German second-tier side Schalke 04, said.
"As long as I feel good and as long as I can help, I'll be here.
"Maybe in the past I could have gone somewhere else and earned more money, but I wanted to play in Europe for as long as I could. I've reached 40, I'm still here, and I'm happy to be able to keep doing it."