Not only would victory give a country that still bears the scars of war its redemptive first-ever berth in an international competition, but it would also come after an atrocious year for football in Bosnia and Herzegovina.
There are no away fans at Zeljeznicar's stadium, nor are there any at any game in Bosnia. A ban was imposed after fierce ethnic fighting at four recent matches, involving all three ethnicities and all three Sarajevo teams.
In Bosnia, until now, most Bosnian Serbs have supported Serbia, most Bosnian Croats have supported Croatia and the fan base for the Bosnian national team has been mostly Muslim, or "Bosniak".
The organisation of the game has until now reflected the legacies of war. The Bosnian football association was run by political appointees unconnected to the game, ensuring it was servile to ethnic interests. These were a Muslim former Minister for Police, a Croat general and a direct appointee of Milorad Dodik, the President of the Bosnian Serb statelet within Bosnia, whose avowed programme is to wreck any state-wide institution.
All this is now changing, however, and Saturday's game puts the changes to the test.
Throughout, the Bosnian national side - and teams like FK Sarajevo - have on the pitch been the country's only functioning multi-ethnic organisations. There is no ethnic veto over a pass down the wing or a cross into the penalty area. Dzeko is a Bosniak Muslim; the national team's most experienced player and co-captain Zvjezdan Misimovic is Serb; the defender Boris Pandza is Croat.
In support of this ground-breaking team, its fans led a revolution against the Bosnian FA.
They organised demonstrations, boycotted matches and staged their own all-star games; they disrupted a game with flares in Oslo for an hour - and they won. In April this year, UEFA and Fifa expelled Bosnia from international competitions until its football association was reformed. Bosnia was readmitted after political appointees were sacked and the association taken over by a "normalisation" committee of sporting figures and heroes.
In parallel, Yugoslav and FK Sarajevo footballing legend Safet Susic was appointed as the team manager, and he in turn enticed back key players who had refused to play under the previous regime.
The fans, the "BH Fanaticos", are led from Bosnia, and include members of the diaspora scattered across the world, refugees who survived massacres, concentration camps and ethnic cleansing.
Their spokesman in the capital, Nizar Smajic, said: "The politicians wanted to impose their interests on our game. We didn't set out to challenge anything political - we just wanted our game back. But politics found us, because politics are everywhere. Now we need to qualify, and the situation will really start to change - we can be rid of all this ethnic shit. If we go to Poland [where the 2012 contest will be staged], we'll take tens of thousands of fans. We took 15,000 to Paris, and most of us missed out on our holidays this year, saving up to go Poland and Ukraine instead."
Such is the excitement in Bosnia that it is infecting the other, Serbian, side of Sarajevo. Mico Simanic - who used to support FK Sarajevo but felt obliged to switch to (Serbian) Slavija after the war - said: "If you told me 10 years ago I'd feel something for the Bosnian national team, I'd have said, as a Serb: 'Never!' But I'll be watching and wanting them to win."
The furthest Zeljeznicar ever got in the UEFA Cup was a semifinal in 1985, losing out on a final against Real Madrid to a late goal by Hungarian side Videoton. The keeper who let it in, Dragan Skrba, a Bosnian Serb who owns the bar, said: "Of course I want Bosnia to qualify - it's just what we need, and Dzeko is the big factor."
There is no overestimating the phenomenon of Dzeko.
Asim Selimovic, who spent his childhood in besieged Srebrenica and was dressed as a girl by his mother to avoid the slaughter in 1995, now lives and studies in St Louis, Missouri. He spoke for fans the world over when he said: "Dzeko is a national idol. He is our pride and joy. When Dzeko scores, every Bosnian refugee in the world has scored. He is our example, our hope."
There is an extraordinary story behind Dzeko's rise - and an extraordinary man who nurtured it. Jiri Plisek, from the Czech Republic, was manager of Zeljeznicar for five months in 2004 and 2005, after which he left "because there were too many interests negatively obstructing the system I wanted."
But Plisek, now manager of FK Sarajevo, "kept my Bosnian heart", he said, and there was one player entirely unappreciated by Zeljeznicar's fans and management whom he desperately wanted to take back to his homeland. "Dzeko had that mental attitude to the game that makes a special player. He needed to apply it, to make up for a lack of tenacity, so I put him in the second team. He was furious, and so were his parents. But it worked - he understood, and showed this inner strength, and the signs of special skill."
Plisek returned to his homeland to take over the Czech side Usti nad Labem, sister team to first-division FK Teplice. He urged Teplice to buy Dzeko for the €25,000 ($43,400) Zeljeznicar wanted for him. "Though it was a pittance, Teplice's attitude was: 'How can good players come from that place down there?' I said that if the club would not buy him, I'd borrow the money myself and do so. That convinced them."
With Dzeko, Teplice won the Czech Cup. In 2007 he moved to Wolfsburg, in Lower Saxony, Germany, for €4 million, where he won a Bundesliga title and became top scorer in the club's history, then to Manchester City for £27 million ($54.5 million).
Yet, unlike others, Dzeko refused offers of both Czech and then German citizenship, which would have taken him to the World Cup finals, choosing Bosnia as his national side - partly his own personal commitment, but also because of the counsel of Plisek, a man of impressively thoughtful modesty.
"These boys reach crossroads where they have to choose who they are, and some understand that glory and money are not everything", he said. "By choosing Bosnia, Dzeko answered that crucial question, 'who am I?', and sent a message to his country, his parents and children. For me, this is how truly great players are made."
At FK Sarajevo, Plisek huddles as a bitter wind blows around the stadium surrounded by the graves of those killed during the siege.
"In this country," said Plisek, "if the politicians had their way, Bosniaks would have to pass to a Bosniak, Serbs to Serbs, Croats to Croats. But that's not how football works; football connects everyone, and has the face of every nationality.
"If they beat Portugal, these Bosnian boys will show the world what their country can do, that their talent can be used in a way the country should follow".
- Observer