Berdych and Almagro received first-round byes, and had contrasting wins.
Almagro, a three-time finalist in Bastad, got off to a rough start against fellow Spaniard Guillermo Garcia-Lopez but rallied to win 3-6, 6-1, 6-1.
His next opponent, Fernando Verdasco, also of Spain, beat Jan-Lennard Struff 6-1, 7-5.
Berdych, the No. 1 seed in the clay-court tournament, easily defeated Argentine Martin Alund 6-3, 6-0. It was the Czech's first match since losing to Novak Djokovic in the Wimbledon quarterfinals.