His most successful tournament was the Hamburg Masters (formerly the German Open), which he won three times (1994, 1995 and 1997).

In the late 1990s, Medvedev's form and results began to flounder until he unexpectedly reached the final of the 1999 French Open where — ranked 100 — he defeated Dinu Pescariu, Pete Sampras, Byron Black, Arnaud Di Pasquale, Gustavo Kuerten and Fernando Meligeni en route.

While their head-to-head record ended deadlocked at 5–5, Bruguera was able to win their two most important matches — the semifinals and quarterfinals of the 1993 and 1994 French Opens, respectively, with Bruguera winning both matches in straight sets.

In the French Open tournament, Medvedev lost six times to the eventual champion (1992–95, 1997 and 1999).

His sister, Natalia Medvedeva, formerly a top 25 player on the WTA Tour, partnered with Andrei to represent Ukraine at the seventh Hopman Cup in 1995, finishing runners-up to Germany's Boris Becker and Anke Huber (Medvedev's girlfriend back then) in the final.