No. 3 seed Bautista Agut won almost three quarters of his service points and saved five break points for victory in 1 hour, 14 minutes.
The Spaniard was leading 4-1 in the first set when play was suspended for 40 minutes due to rain.
Fognini, who defeated Germany's Philipp Kohlschreiber in last year's final, had been bidding to reach his fourth final of the season.
Rosol struck 10 aces to Youzhny's five, and converted four of his nine break points to win in just under 1½ hours. Youzhny won the tournament in 2002.
Rosol upset the fourth-seeded Feliciano Lopez of Spain in the quarterfinals.
Both the 48th-ranked Czech and Bautista Agut are aiming for their second title. Rosol won at Bucharest last year while the 23rd-ranked Spaniard's maiden victory came at `s-Hertogenbosch last month.