Medvedev, 26, took two hours 35 minutes to beat Serbian Djere in a men's singles match at Court Philippe Chatrier. The second-seeded Russian will play No. 28 Miomir Kecmanovic of Serbia for a berth in the fourth round.
Medvedev made the quarter-finals in 2021.
Top seed and hot favourite Novak Djokovic of Serbia and Spaniards, Rafael Nadal and and Carlos Alcaraz have already entered the third round.
No. 8 Casper Ruud also reached the third round at Roland Garros. The 23-year-old Norwegian beat Emil Ruusuvuori of Finland 6-3, 6-4, 6-2 on Thursday.
Ruud won the Geneva Open last Saturday for his seventh career title on clay courts.
Earlier in the day, No. 3 Paula Badosa became the first woman seeded in the top 10 to reach the third round.
She defeated 68th-ranked Kaja Juvan Slovenia 7-5, 3-6, 6-2 at Court Suzanne Lenglen in more than two hours.
Badosa, a quarter-finalist at the 2021 French Open, will face No. 29 Veronika Kudermetova next.
But two-time Grand Slam finalist Karolina Pliskova lost her second round to wild card Leolia Jeanjean’s 6-2, 6-2.
She is the latest top 10 in women's singles to crash out. Others were No. 2 Barbora Krejcikova, No. 4 Maria Sakkari, No. 5 Anett Kontaveit, No. 6 Ons Jabeur and No. 10 Garbine Muguruza.
The remaining four in the top 10 are: No. 1 Iga Swiatek, No. 3 Paola Badosa, No. 7 Aryna Sabalenka and No. 9 Danielle Collins.