The Swiss world number one was given a stern test by Colombian Alejandro Falla - who almost stunned him at Wimbledon in 2010 - before winning 6-3 5-7 6-3.
American fourth seed Williams beat Jelena Jankovic of Serbia 6-3 6-1.
Belgium's Kim Clijsters, set to retire at the end of the season, overcame Roberta Vinci of Italy 6-1 6-4.
Clijsters next plays Spaniard Carla Suarez Navarro, while Williams will take on Urszula Radwanska of Poland.
"It's a huge deal," said Federer about the prospect of winning his first Olympic singles gold medal. "It would be a massive highlight of my life.
"I remember the emotions I had winning the gold with my Stan [Wawrinka, in the doubles at Beijing 2008].
"I have a chance, I'm the number one player in the world so I should have a chance, at medals and hopefully gold.
"It's already the fourth Olympics for me so I will give it all I have."
Czech sixth seed Tomas Berdych, the 2010 Wimbledon runner-up, was upset 6-4 6-4 by Steve Darcis of Belgium.
Berdych was a potential quarter-final opponent for Britain's Andy Murray, who faces Stanislas Wawrinka on Sunday.
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