Szavay broke early in both sets, but couldn't capitalize. In the opener, the Hungarian had a chance to serve out the set when leading 5-4, but she hit two double-faults and made two bad misses with her forehand.
Dushevina, a Russian who reached last year's final, eventually won a tight tiebreaker to reach the next round.
Radwanska fell behind 0-40 in the opening game, but the 10th-ranked Pole rallied to win it and was never challenged the rest of the way.
"I'm very pleased with the way I'm playing right now," Radwanska said. "I haven't played a tournament since Wimbledon. I hope to reach the final here because this event is good preparation for the Olympics."
Pavlychenkova, a former top-ranked junior from Russia, trailed 4-1 in the second set before getting to 4-4. But Srebotnik, a Slovenian who won one of her four WTA titles in Stockholm three years ago, broke back to lead 5-4 and served out the match.
"She played well for a while and I started missing a few shots," Srebotnik said. "She's a very good player but there's a long way to go from junior tennis."
Srebotnik, ranked 29th, relied on her strong first serve. Pavlychenkova only managed to take five of 31 points when Srebotnik put her first serve in.
Sixth-seeded Sania Mirza of India ousted Emilie Loit of France 6-3, 6-2 in the first round. Mirza's match was moved to Wednesday because she reached the doubles semifinals at a WTA event at Carson, California, last weekend.
"I felt very tired and jet-lagged and expected a tough match," Mirza said. "It was nice to win in straight sets."