"I think everybody in Bosnia is still awake and watching me, which makes me really proud for my country," Dzumhur said in comments published on the ATP website.
"For such a small country as Bosnia, it's a really big result.
"It's going to be a big thing for me, for my country, as the only player in the Top 100 making great results.
"So I'm proud of my country and proud of myself and it's going to be a good day tomorrow."
Dzumhur, who is ranked 67th, is not the first Bosnian-born player to make an ATP Tour final, with others to have done so under the flag of another country.
The 2014 U.S. Open winner Marin Cilic played for Bosnia as a junior but has represented Croatia as a professional.
Ivan Dodig, a 2015 French Open doubles winner, played for Bosnia until 2008 before also shifting allegiance to Croatia. He won his only ATP Tour singles title in 2011.
Against Edmund, Dzumhur had to fight off the lethargy of playing 11 matches in 12 days, having reached the final of a Challenger Tour event in the Dominican Republic last week.
He did not arrive in North Carolina until the early hours of Monday and was back on court later in the day.
He looked headed for an earlier departure for the U.S. Open that starts Monday, after being routed in the first set.
But he rallied when the adrenalin kicked in.
"I was a little bit tired," Dzumhur added. "In the second set I tried to hold my serve and I broke him in a game when he was 40-15 up, and in that moment everything changed.
"As the adrenaline comes up you don't feel (tired) anymore. I think I will be ready for the final. It's my first final, so I have to be."
Bautista Agut easily accounted for Germany's Jan-Lennard Struff 6-2 6-4 in the other semi-final and has a chance to make amends for last year's loss in the title-decider to fellow Spaniard Pablo Carreno Busta.
(Reporting by Andrew Both in Cary, North Carolina; Editing by Ian Ransom)
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