The pair will meet in the next round after top seed Youzhny saw off France's Kenny De Schepper 6-3 6-4, while Haase - seeded seventh - edged his contest with plucky wildcard Henri Laaksonen 6-4 1-6 7-5.
Defending champion Youzhny was rarely troubled in his straight-sets win over De Schepper - progressing in just over an hour at the ATP 250 clay-court event.
In the pair's first meeting, the world number 19 showed his quality to get the crucial break of serve in each set and win through, even though De Schepper hit seven aces and save five of seven break points.
With Youzhny having ensured his quarter-final berth, next it was Haase's turn - and he did not disappoint, despite being severely tested by world number 319 Laaksonen.
Haase fought back from two games down against the 22-year-old to take the first set before Laaksonen stormed back to take the second in style.
However, the Dutchman's quality showed in the end as he edged a closely fought decider - saving four of six break points in the final 66 minutes of an enthralling affair.
While Haase joined Youzhny in the next round, fourth seed Fernando Verdasco and Jan-Lennard Struff must continue their match on Friday after rain interrupted play.
The pair are level on one set apiece and 1-1 in the decider.
Either Verdasco or Struff will meet Viktor Troicki in quarters after he dispatched Kazakh opponent Andrey Golubev.
The Serb returned from his year-long doping ban with victory over Dominic Thiem on Monday and continued his strong form against world number 59 Golubev.
Hitting seven aces and saving six of seven break points, the wildcard beat Golubev 6-3 6-4 in 67 minutes.