Djokovic struggles into quarters
Djokovic was made to work hard for his win over Andreev
The Serbian looked to be struggling physically when pegged back to one set all but found his form in the decider.
He faces Nicolas Almagro next, after 12th seed Fernando Gonzalez pulled out of their match with a leg injury.
Djokovic remains on course for a semi-final showdown with Roger Federer, who plays Ivo Karlovic later on Thursday.
“I’m very relieved and satisfied to get through this one,” said Djokovic. “He’s one of the most difficult players to beat on this surface.
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“I was trying to be aggressive and I think the patience was very crucial today.”
After suffering with illness at Monte Carlo two weeks ago, and despite looking uncomfortable early on against Andreev, he added: “I didn’t know what to expect from my physical condition and I’m very happy.
“I’m a little bit tired but not as much in the last few weeks and recent matches.”
Andy Roddick and Nikolay Davydenko are in action in the other half of the draw, which is without Rafael Nadal following the defending champion’s surprise loss to Juan Carlos Ferrero on Wednesday.