The American world No.2 must have thought she would have an easy morning after powering through the first set in sensational fashion but the Russian bounced back in the second to test her opponent.
"The whole match I played well. She stepped up her game in the second set. Overall I thought I played great," Gauff said.
"I move well on clay; really comfortable with sliding and moving on the surface. The most physical surface for sure and I do well in that department."
Hunting her first French Open crown after reaching the final in 2022 and semi-finals last year, the 21-year-old started fast, earning three consecutive breaks for a 5-0 lead in just 15 minutes.
Gauff was running her opponent ragged across the baseline, with the 30-year-old Alexandrova, bidding to reach her first grand slam quarter-final, earning a mere five points until that stage.
The American, who has now won four of the pair's five meetings, gave away five break points in the next game but still secured her first bagel of the tournament before the Russian got on the scoreboard at the start of the second set with her first hold.
Unforced errors started creeping into Gauff's game as Alexandrova put up stronger resistance.
Gauff, the youngest American to have reached at least the fourth round at seven consecutive grand slams since Venus Williams between 1997-1999, broke Alexandrova at 3-3.
The Russian broke straight back and went 5-4 up, with Gauff clearly rattled and double-faulting twice before holding to level.
But the second seed kept her composure, broke Alexandrova and wrapped up the match on her serve. She will next face the winner of the all-American fourth-round clash between Madison Keys and Hailey Baptiste.
With agencies
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