Serena will next play No. 13-seeded Vera Zvonareva of Russia. Williams won when they met for the 2010 Wimbledon title.
"It's not an easy tournament, playing someone tough so soon," Williams said.
Roddick converted only two of 13 break-point chances against Klizan but dominated with his serve, as often happens at Wimbledon, where he's a three-time runner-up.
"I served really well," he said. "I would have liked to have converted more break points. But I got a couple and, thankfully, that's all I needed."
The Olympic match, delayed a day by rain, was Roddick's first in eight years. He lost in the third round at the 2004 Games and skipped Beijing in 2008.
Roddick, a dedicated Davis Cup player for the United States, took Court 2 on a sunny morning looking particularly patriotic. He wore a blue shirt, white shorts and stars-and-stripes shoes.
"Posh sneakers, Andy," shouted a spectator with a British accent.
He lost only nine points in 11 service games, and the first break point came on the final shot of the opening set, which put him ahead to stay.
Roddick won with more than just his serve, hitting several improbable shots, including a back-to-the-net lob to help win one point, and a backhand as he did a belly flop behind the baseline.
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