His serve broken three times in the second set, Djokovic came out fighting in the third, and the No. 4-ranked Serb strolled to victory over a weakened Verdasco, who had 19 unforced errors for the match.
"I had ups and downs in today's match," said Djokovic. "Verdasco used his chances in the second set when he broke."
"I saved energy at the end of the second set for the refreshing start of the third and I was fortunate to do so."
Djokovic's semifinal opponent will be the winner of Friday's later quarter-finals between Croatia's Ivan Ljubicic and Sweden's Robin Soderling.
Svetlana Kuznetsova became the first woman to qualify for the semi-finals, dispatching fellow Russian Anastasia Pavlyuchenkova 6-3, 6-3.
Sixth-ranked Kuznetsova faced little challenge overcoming No. 39 Pavlyuchenkova, who on Monday defeated Venus Williams for the second time in a week. Kuznetsova will face the winner of Friday's second quarterfinal between Russia's Nadia Petrova and China's Peng Shuai.
In later Friday action, No. 2-ranked Rafael Nadal faces Russia's Marat Safin and Ljubicic plays Soderling. In the women's draw, Elena Dementieva of Russia meets Poland's Agnieszka Radwanska and Marion Bartoli of France plays Russia's Vera Zvonareva.