Klizan hit five aces and never faced a break point. He will next play Ricardas Berankis of Lithuania, who advanced when eighth-seeded Jurgen Zopp of Estonia retired with a back injury after losing the first set 6-3.
Earlier, fourth-seeded Fabio Fognini eliminated 2009 champion Serghiy Stakhovsky of Ukraine 7-6 (4), 6-4, while fellow Italian Flavio Cipolla made the final eight with a 6-2, 6-7 (5), 6-3 victory over Grega Zemlja of Slovenia.
Youzhny, the 2004 champion and a two-time runner-up, served hard in the first set and allowed Ram to score only three points on receiving. Youzhny then made a break in the opening game of the second set and held his serve to close the match on his first match point.
"I've opened really strongly and it was enough to win the set," Youzhny said. "It was much easier in the second set after I made a break."
In his eighth quarterfinal appearance at the tournament since 2001, Youzhny on Friday will face Guillermo Garcia-Lopez of Spain, who was leading 6-1 and 3-1 when Paolo Lorenzi of Italy retired with a left Achilles injury.
Fognini was leading 3-1 and 40-15 in the first set when he made three double faults and Stakhovsky broke him and forced a tiebreaker.
"The tiebreak was really important and I played great the first points," Fognini said. "I went 6-1 and then 6-4."
He then accelerated to a 5-1 lead in the second set and though the Ukrainian broke for 5-4, Fognini served the match out to love.
He will next face Roberto Bautista-Agut of Spain, who ousted Russia's Teimuraz Gabashvili 6-3, 6-4, while Cipolla will play against Daniel Gimeno-Traver of Spain, who stopped Lukasz Kubot of Poland, 6-4, 7-6 (5).