Not that this Grand Slam run is being greeted with the kind of hullabaloo that accompanied her journey to last year's U.S. Open final in which she lost to compatriot Sloane Stephens.
"Hopefully many more to come", he said in his post-match on-court interview.
American Stephens will face the victor of second seed Caroline Wozniacki's fourth-round match against Russia's Daria Kasatkina, which takes place later on Sunday.
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.
The 10th-seeded Stephens was a 6-2, 6-0 victor against No. 25 Anett Kontaveit of Estonia in 52 minutes, while No. 13 Keys beat No. 31 Mihaela Buzarnescu of Romania 6-1, 6-4 in 65 minutes.
Both Keys and Stephens have now managed to complete a set of quarterfinal runs at all four Grand Slam sites.
With a 7-5, 4-6, 6-0, 6-3 upset victory against eighth-seeded David Goffin of Belgium on Sunday, the 72nd-ranked Italian became the lowest-ranked man to reach the quarterfinals at Roland Garros since Ernests Gulbis, then ranked 80th, in 2008.
Seconds later, she threw her head back after acing Buzarnescu for the win.
Novak Djokovic advanced to the fourth round of the French Open by beating Roberto Bautista Agut 6-4, 6-7 (6), 7-6 (4), 6-2. Today, she clinched the 21st Grand Slam match win and extended to 12-4 her win-loss record in Paris.
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Rather than thinking about what might happen a few days down the line, Stephens was in a reflective mood after finally getting over her fourth-round hump in Paris.
However, Keys faced a set point at 5-6 and needed some superb defence to fend off the fired-up Japanese. He'll play the second-seeded Alexander Zverev next.
Earlier, Japan's Naomi Osaka's bid to reach the fourth round at Roland Garros died in a straight-sets defeat to her American nemesis Madison Keys.
The 23-year-old Keys will now be eyeing a semi-final spot, with Czech 26th seed Strycova or Putintseva waiting in the quarter-finals.
He shrugged off a question about spending so much time on court, saying: "I'm young so I may as well stay on court to practice a little bit".
On a sunny Court Philippe Chatrier, the 13th-seeded American had two match points at 5-2 and 40-15.
Serving a second time for the match, the 23-year-old Keys let out a scream of "Come on!" when she hit a superb cross-court backhand victor to make it 40-15 and give herself another two chances for victory.
A brilliant reflex backhand victor was another Buzarnescu highlight as she reduced the deficit with another hold - but Keys, going back to basics, still had one more opportunity to serve the match out. The 28-year-old, seeded 19 in his first Grand Slam since Wimbledon last year after struggling with injury, saw off Simon 6-3, 6-1, 6-3.