Four-times champion Rafael Nadal, however, was still too strong for Australian Lleyton Hewitt, who proved a tough nut to crack but lost 6-3, 6-4, 6-3.
"It was hard to play today. There was a lot of wind and some rain at times," said Nadal.
"It was a surprise that Juan Carlos lost, and also I think David played in a kamikaze way, and it was difficult for him to stop on time. David played too flat."
Verdasco, the seventh seed, went to the brink against Germany's Philipp Kohlschreiber but he refused to let go, prevailing 2-6, 6-3, 6-3, 6-7, 6-4 despite a foot injury.
"I broke my big toenail and it did hurt like hell, I was almost crying," said Verdasco.
Almagro, the 19th seed, eased past Ukraine's Alexandr Dolgopolov 6-3 6-3 6-4.
Fernando Verdasco reacts after winning his match against Philipp Kohlsschreiber
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