Swiatek is the dominant player on the WTA Tour after success on clay and hard courts, but has still not got to grips with the Wimbledon grass.
The five-time major winner is known for her clinical brand of tennis, but she hit a rare 34 unforced errors to Putintseva’s 18.
The Kazakh, cheered on by a raucous crowd on Court One, wrapped up the victory on her third match point.
Putintseva has reached the quarter-finals of the French Open and US Open, but had never been beyond the second round at Wimbledon.
It continues a superb grass-court season for the 29-year-old, who won the Birmingham Classic last month.
She will face 13th seed Jelena Ostapenko next after the Latvian claimed a commanding 6-3 6-1 win over American Bernarda Pera.
Meanwhile, grass remains Swiatek’s least successful surface, with Wimbledon being the only Grand Slam in which she has not gone past the quarter-finals.
The 23-year-old French Open champion will return to her favoured clay courts to prepare for tennis at this summer’s Olympics, which will be hosted at Roland Garros.
