Mannarino celebrated his 30th birthday on Friday by overcoming fourth-seeded Gael Monfils 4-6, 7-6 (4), 6-4 in an all-French contest to reach the final for the second straight year.
He seeks his first career title in his fifth ATP final.
Monfils served 17 aces to Mannarino's one, but converted only two of his seven break chances. Mannarino converted two of his four.
Dzumhur, of Bosnia and Herzegovina, beat Jiri Vesely of the Czech Republic 6-3, 7-6 (1) in the first semifinal.
Dzumhur saved two set points at 4-5 in the second set.
He has won two ATP titles, both last year in Russia.
"It's the first time I have played two tournaments on grass prior to Wimbledon," Dzumhur said. "So I have never given myself the chance to play well on grass. I did feel like I could play well, but I needed matches."