World number 98 Putintseva showed off her wide range of shot-making skills to break Strycova twice in the early exchanges before serving out the opening set in 52 minutes.
Strycova, who was making her first fourth-round appearance in Paris, struggled to find her rhythm and continued to spray the court with unforced errors as Putintseva raced to a 4-1 lead in the second set.
The 23-year-old dropped her serve in the next game but did enough to see off a late challenge from Strycova and wrap up the victory on Court Suzanne-Lenglen.
Putintseva, who also defeated British number one Johanna Konta in the opening round, has matched her best performance at Roland Garros by reaching the quarter-finals for the first time since 2016.
"Today I was very tough mentally," she said after the match.
"It was a fourth-round match and everyone is play good now. You just have to fight for every point."
Up next for Putintseva is U.S. Open runner-up and 13th seed Madison Keys, who is yet to drop a single set over four matches in the tournament.
(Reporting by Hardik Vyas in Bengaluru)
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