With sweltering temperatures reaching a Wimbledon record 35.7C, five-time champion Williams kept her cool to dispatch world number 93 Babos with 12 aces and 23 winners in just 59 minutes on Centre Court.
The 33-year-old, bidding for a third Grand Slam title of the year, will play Britain's Heather Watson for a place in the last 16.
"I'm definitely playing well, which I never say, but I had two really tough matches straight off the bat so I had to pick up my game," Serena said.
"I really had no option but to play well so far. But I'd like to believe I have a lot of room to improve."
Also Read
Williams has swept to Australian and French Open triumphs over the last six months, taking her within two major titles of becoming the first woman since Steffi Graf in 1988 to claim the four top titles in one year.
The American great also won the US Open to end 2014 on a high and is heavy favourite to continue that dominant run by taking the Wimbledon title for the sixth time in her illustrious career.
Her failure to make the All England Club quarter-finals for the last two years led Serena to caution against labelling her as a certainty to become the first woman to win the French Open and Wimbledon back to back since she achieved the feat in 2002.
But, on this evidence, she is approaching peak form at just the right time.
That resounding statement of intent was enough to seal the first set and she raced through the second in even quicker time to complete a comfortable stroll.