The top-seeded Russian saved six match points in the final game before finally losing.
Haase, a Dutchman seeded seventh, will play Juan Monaco in Saturday's semifinals. Monaco beat two-time champion Thomaz Bellucci of Brazil 7-6 (1), 6-1.
In a match held over from Thursday, fourth-seeded Fernando Verdasco of Spain completed a 3-6, 6-3, 7-6 (3) win over Jan-Lennard Struff of Germany.
Verdasco returned on court later to beat Viktor Troicki of Serbia 6-4, 6-7 (7), 6-1 in the last quarterfinal match.
Verdasco set up an all-Spanish semifinal against Pablo Andujar, who beat another Spaniard, second-seeded Marcel Granollers, 7-5, 6-3.