The 26-year-old Argentine broke the 96th-ranked Vliegen to take the first set. Nalbandian, the 2002 champion and twice runner-up since, lost his first match point at 6-5 in the final set when Vliegen hit a running backhand winner from wide of the court.
He clinched when Vliegen failed to control a strong service return.
Juan Martin Del Potro, the No. 3 seed, made just four unforced errors as he cruised past Switzerland's Stephane Bohli 6-3, 6-3.
The Argentina Davis Cup teammates are on course to meet in a semifinal Saturday.
Fourth-seeded James Blake of the United States had 18 aces as he rallied from a set down to beat 76th-ranked Oscar Hernandez of Spain 6-7 (4), 6-2, 6-4.
"I was serving well and that got me out of a lot of trouble,'' Blake said.
Simone Bolelli of Italy beat Marcel Granollers of Spain 6-4, 6-2. In the quarterfinals he will face either top-seeded Roger Federer or Jarkko Nieminen of Finland, who play later Thursday.
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