The defending champion started Friday's match at the ATP 250 clay-court tournament in Gstaad well however he was unable to cope with a resurgent Haase who claimed a 3-6 6-1 6-4 win in just under two hours.
Having broken his opponent twice in the opening set, the Russian eventually moved ahead having converted his sixth set point.
That should have been a base to build on for the top seed but all his momentum disappeared as Haase won five consecutive games at the start of the second.
Haase, the seventh seed, won 84 per cent of points on his own serve in the second set, while taking 55 per cent against the normally reliable Youzhny serve.
With the match going into a decider, it was the Dutchman who took the initiative and, although Youzhny saved six match points, Haase prevailed to set up a semi-final with Juan Monaco.
The Argentine saw off the challenge of another former Swiss Open winner as he beat Thomaz Bellucci 7-6 (7-1) 6-1.
Prior to the quarter-finals, there were three former winners in contention but Marcel Granollers' exit to Pablo Andujar ensured none remain and there will be a new champion in 2014.
A five-game winning run saw Andujar claim the opening set and start the second well, and he went on to brush aside the second seed 7-5 6-3.
The Spaniard will face compatriot Fernando Verdasco after he completed two victories on Friday.
Having beaten Jan-Lennard Struff 3-6 6-3 7-6 (7-3) in their delayed second round match the fourth seed also went on to beat Viktor Troicki in the quarter finals.
Verdasco ended the run of the Serbian, continuing his comeback from a year-long doping ban, in two hours four minutes to reach the last four with a 6-4 6-7 (7-9) 6-1 win.