The 24th-seeded Monfils, who reached the quarterfinals at the French Open earlier this month, has never advanced past the third round at the All England Club on a surface he struggles with.
Monfils hit 71 winners, including 23 aces, but had trouble returning his opponent's serve.
Vesely, a former junior world No. 1, is playing at Wimbledon for the first time. The Czech could face another Frenchman in the next round, 13th-seeded Richard Gasquet.