Alcaraz, who has been nursing an arm injury, fought hard to win 6-4 7-6 (7-5) 6-3 at Roland Garros.
The reigning Wimbledon champion had enough quality to come through an entertaining night-session match.
“It was a really good match, I think I played really well – much better than my previous matches,” said 21-year-old Alcaraz.
“I wanted feel myself on court – I think I did that pretty well.”
Alcaraz pounced to decisively break Korda’s serve for 5-4 in the opening set and confidently held to move in front.
The second set was equally tight. The pair exchanged breaks early on before Alcaraz clinched the tie-break at the second attempt with a blistering forehand winner.
With Alcaraz’s confidence and quality continuing to rise, the third set started to run away from 23-year-old Korda.
Alcaraz broke for 3-1 and made Korda dig in to save two more break points at 4-1 before the two-time major winner served out an encouraging victory.
Alcaraz, who reached the semi-finals last year in a career-best run, will face American 15th seed Ben Shelton or Canadian 21st seed Felix Auger-Aliassime in the fourth round.
Having grown up on clay courts in his home region of Murcia, Alcaraz is considered by many as the heir apparent to Rafael Nadal at Roland Garros and it would be a surprise if his talent does not lead to a triumph here.
Whether that will be this year, after a injury-hit build-up to the major, remains to be seen – but this victory was a step in the right direction.
Alcaraz had not played for a month because of the pain in his right forearm, missing the Italian Open as a consequence and arriving in Paris with only a run to the Madrid Open quarter-finals under his belt on clay.
After opening with a straight-set win over American lucky loser JJ Wolf, Alcaraz was nowhere near his best in his second-round match against Dutch qualifier Jesper de Jong.
Korda was another step up in class, because while most of his success has come on hard courts, the American has proved to be adept on clay and beat Alcaraz in Monte Carlo two years ago.
Alcaraz avenged that defeat a few weeks later when they met in the French Open third round – and inflicted the same outcome at the same place this year.
“It is difficult to play against him and I had to run a lot. It was like a marathon for me. But I wasn’t worried. I know what I’m capable of.”
