Spain’s world number 38 lost the first set to Filippo Volandri but came back to win 4-6 7-5 6-2 and set up a last-eight clash with Lukas Rosol.
Czech Rosol beat Romania's Victor Hanescu 7-6 6-3 in a late match in the Serbian capital.
Andujar needed two hours and 20 minutes to beat Italian Volandri, and said it was a tough task.
"It was a difficult match for me," admitted Andujar. "The conditions weren’t easy; the court was very fast and it was hot, which made it a little bit more difficult.
"Always in your first match you have to try and stay as focused as you can; it’s never going to be easy. I was down 6-4 5-4 and I tried to continue playing my game and finally I made it and I’m really happy for that."
Brazil’s Joao Souza booked a quarter-final date with Argentine David Nalbandian after his Serbian opponent Dusan Lajovic retired at 4-2 down in the first set.
Frenchman Benoit Paire surprised Spain’s Guillermo Garcia-Lopez in a three-set epic, winning 4-6 7-5 7-5 to set up a clash with one of Jarkko Nieminen or Antonio Veic, whose match is poised at 5-3 in the Finn's favour in the first set and will be played to a conclusion on Friday.
