Djokovic, the 2008 champion, had to come from a break down in each set against the Brazilian, who put up a lively display but at times had trouble keeping his big serve under control.
Verdasco, fresh from his triumph at last week's Barcelona Open, booked his place in the last eight with a 6-4, 7-6 win over his compatriot Guillermo Garcia-Lopez.
Wozniacki crashes out
Caroline Wozniacki's ankle injury returned to haunt her as the top seed was knocked out of the Stuttgart Open in straight sets by Lucie Safarova.
The world number two, who retired hurt during a match at the Family Circle Cup in Charleston two weeks ago after injuring her ankle in a tumble, said she was still struggling as she lost 6-4, 6-4 in her first match in Stuttgart. On Wednesday, four-times French Open champion Justine Henin overcame a broken finger and fired-up German opponent to win her opening match her first clay court tournament since her comeback.
The 27-year-old was taken to a tiebreak in the first set before breaking Julia Goerges' resistance in the second for a 7-6, 6-1 first-round win.
Li Na of China got the netter of third seeded Russian Svetlana Kuznetsova 6-3, 7-5 while Agnieszka Radwanska (5) went out to Israel's Shahar Peer 6-3, 6-7, 6-2.
Victoria Azarenka (6) of Belarus eased past Italian Flavia Penneta 6-1, 6-4 with Serbia’s Jelena Jankovic dispatching Gisela Dulko of Argentina 6-2, 6-2.