The fifth-seeded Nadal closed out Verdasco in 89 minutes while persevering through 95-degree heat.
The ninth-seeded Federer faces American Steve Johnson on Wednesday night.
Federer defeated Nadal in a memorable five-set Australian Open final earlier this year.
Verdasco put up a strong challenge in the second set but the Spaniard was stymied by his countrymate. Nadal broke Verdasco's service to take a 6-5 lead and then finished the match.
Fourth-seeded Kei Nishikori of Japan also advanced with a 6-2, 6-2 victory over Gilles Muller of Luxembourg.
Nishikori next faces American Donald Young, who upset 14th-seeded Lucas Pouille of France 6-4, 1-6, 6-3. Young won the match in one hour, 47 minutes.
In another match, Tunisia's Malek Jaziri was a 6-4, 3-6, 6-3 victory over American Taylor Fritz.