Konta, who before this year had never won a match at the claycourt major in four previous appearances, struck 33 winners, including seven aces to dispatch the 23rd-seeded Croatian on the hottest day of the tournament so far, with temperatures set to rise to as high as 31 degrees Celsius.
Vekic was broken five times.
The last British woman to reach the Paris quarters was Jo Durie in 1983. Durie eventually lost in the semis.
Although Konta and Vekic had split their previous six encounters, including an epic clash in the second round of Wimbledon in 2017 that lasted more than three hours, this was their first meeting on clay.
After three straight breaks of serve, Konta was the first to hold, going up 3-1.
With Vekic struggling to land a first serve in, Konta went up a double break for 5-2. Serving for the first set, the Briton saved four break points before taking it with her third ace.
After swapping breaks early in the second set, a drop shot handed Konta another break. She held serve to love clinch the match in 84 minutes.
Konta will play either 2017 U.S. Open winner Sloane Stephens or twice Grand Slam champion Garbine Muguruza for a spot in the semi-finals.
(Editing by Pritha Sarkar)
