Verdasco, who pushed Wimbledon winner Andy Murray to five sets in the quarterfinals of that tournament, hit 12 aces as he eliminated Sweden's Andreas Vinciguerra 6-4, 7-6 (3). Vinciguerra was the last home player remaining in the clay-court ATP tournament in Bastad.
Troicki easily beat Joao Souza of Brazil 6-4, 6-2, while Italy's Filippo Volandri routed Estonia's Jurgen Zopp 6-1, 6-2.
Thiemo De Bakker, Albert Ramos, Blaz Kavic, Henri Laaksonen, Jan-Lennard Struff and Guillermo Garcia-Lopez also advanced.
Results from the second day of the ATP Swedish Open (x denotes seeding):
First round:
Filippo Volandri (ITA) bt Jurgen Zopp (EST) 6-1, 6-2
Jan-Lennard Struff (GER) bt Sergiy Stakhovsky (UKR) 6-3, 6-3
Guillermo Garca-Lpez (ESP) bt Jan Hjek (CZE) 7-6 (7/4), 6-1
Viktor Troicki (SRB x6) bt Joo Souza (BRA) 6-4, 6-2
Albert Ramos (ESP) bt Julian Reister (GER) 6-3, 6-3
Blaz Kavcic (SLO) bt Marius Copil (ROM) 6-4, 6-1
Thiemo de Bakker (NED) bt Diego Schwartzman (ARG) 6-2, 6-4
Filippo Volandri (ITA) bt Jurgen Zopp (EST) 6-1, 6-2
Henri Laaksonen (SUI) bt Antonio Veic (CRO) 5-7, 6-1, 6-1