The 26-year-old Spaniard was dominant throughout the 6-1, 6-2 win over Djokovic, the 2008 Australian Open winner.
Verdasco, who lost the Australian Open semifinal to eventual champion Rafael Nadal in five sets last year, will next face the winner of Friday's semifinal between Frenchman Jo-Wilfred Tsonga and U.S. Open champion Juan Martin Del Potro.
The Australian Open starts Monday in Melbourne.
Earlier Thursday, Chile's Fernando Gonzalez beat German Tommy Haas 6-2 6-4.
Del Potro has wrist injury, pulls out of KooyongLa Liga round-up