Second seed Williams was broken three times in the opening set at Rod Laver Arena and again when coasting to the finish line in the second but shifted up a gear when it counted to close out the match in one hour and 46 minutes.
"I think she's a really smart player, she can do everything," said Williams of the hard-running Strycova, who caused at shock at last year's tournament by dumping third seed Garbine Muguruza out in the third round.
"It's always good I have something I can improve on and I know I can do better on my serve."
Williams, bidding to win a record 23rd grand slam title in the professional era, has now notched 10 consecutive quarter-final appearances at the majors, dating back to the 2014 US Open.
The 35-year-old great next faces the winner of the match between Russian Ekaterina Makarova and British ninth seed Johanna Konta.
Williams came into the tournament with doubts about her fitness after she had only two tour matches of preparation since the US Open, but she is yet to drop a set at Melbourne Park.
