The 22-year-old Italian, ranked No. 67, last reached the final four in July 2008 at Umga, Slovenia. He is looking for his first title on the ATP tour.
Fognini's match was interrupted by rain for about 30 minutes at 4-4 in the third set.
Davydenko double-faulted to give two match points to Fognini, who won when the Russian sent a backhand wide on the second.
The rain returned after the match and the organizers were forced to postpone the other two quarterfinals until Saturday.