Nadal fell behind 2-0 against the 99th-ranked Bagnis of Argentina, then cleaned up his act and took 14 of the next 15 games Thursday en route to winning 6-3, 6-0, 6-3.
A record nine-time champion at Roland Garros, Nadal has won 200 of 230 matches in majors, a winning percentage of .870. Only seven other men have earned that many victories at Grand Slam tournaments; Roger Federer, who is missing from this year's French Open because of injury, holds the record of 302.
The fourth-seeded Nadal owns 14 Grand Slam titles -- tied with Pete Sampras for second most behind Federer's 17 -- including a record nine at Roland Garros.
Nadal may have reached his milestone, but two others are also attempting to add to their résumé Thursday.
Novak Djokovic, the current top-ranked player, can win his 50th match at Roland Garros by defeating qualifier Steve Darcis of Belgium. But the milestone that really matters to Djokovic is winning his first French Open, the only Grand Slam he hasn't won.
The last Tunisian to reach the third round at Roland Garros was Mustafa Belkhodjia in 1963.
Berdych, a 2010 Wimbledon finalist and French Open semifinalist, says he was "really pushed, and I was able to come back and win."
The Associated Press contributed to this report.