The US Open champion, who reached the quarterfinals at Roland Garros last year after four consecutive first-round exits, played nearly flawless service games and broke Kecmanovic twice in the opening set before another early break gave the Russian player a two-set lead.
"It was magnificent today. I did not have my serve broken," Medvedev, the runner-up at the Australian Open this season, said in an on-court interview. "On this surface it is quite unusual. Hopefully there are more such matches for me next week."
Unlike the women's draw, where No. 1 Iga Swiatek is the only top-10 player remaining, the relatively surprise-free men's results continued to pour in, with Medvedev, No. 4 Stefanos Tsitsipas, No. 7 Andrey Rublev, No. 8 Casper Ruud, No. 11 Jannik Sinner, No. 12 Hubert Hurkacz and No. 20 Marin Cilic advancing. All won in straight sets, except for Rublev, who required four, and Ruud, who prevailed in five sets.
Medvedev arrived at the French Open with only one match on clay under his belt after undergoing a procedure to treat a hernia in April. He finished the contest on his first match point, firing a cross-court forehand past Kecmanovic.
He next plays Cilic, who defeated Gilles Simon 6-0, 6-3, 6-2.
Sinner saved 11 set points in the second set on the way to beating Mackenzie McDonald 6-3, 7-6 (6), 6-3 in the third round at Roland Garros.
A 20-year-old from Italy, Sinner has made it to Week 2 in all three of his trips to the French Open. He wound up losing to Rafael Nadal in 2020 and 2021.
McDonald is a Californian who is No. 60 in the ATP rankings and was playing in the third round in Paris for the first time. He fell to 1-16 against opponents ranked in the top 15.
Sinner's next opponent is Rublev, who needed four match points to get past Cristian Garin 6-4, 3-6, 6-2, 7-6 (11).
In the final match of the day, teenager Holger Rune has given Denmark a man in the fourth round at Roland Garros for the first time in 63 years.
The 19-year-old Rune, who won the French Open junior title three years ago, closed out the third round of the main draw by beating Hugo Gaston of France 6-3, 6-3, 6-3 on Saturday night. The 40th-ranked Rune has not dropped a set so far in the tournament after never having won a Grand Slam match until beating 14th-seeded Denis Shapovalov in his Roland Garros debut this week.
Rune will face the fourth-seeded Tsitsipas in the fourth round.
Reuters and The Associated Press contributed to this report.