In January 2010, it was announced that Antonio signed with Arequipa-based side FBC Melgar for the start of the 2010 Descentralizado season.
[1] Meza Cuadra made his debut for FBC Melgar in the Peruvian First Division on February 14, 2010, in a home match against Club Juan Aurich.
[3] Meza Cuadra played one of his best games of the 2011 season in round 29 at home against Alianza Lima, the league leaders at the time.
During the match, Alianza Lima took the lead twice, but then Meza Cuadra managed to score in the 27th minute and cross the ball that led to Christian Ramos' own goal.
The match finished in a 3–2 win over Alianza Lima and guaranteed Melgar's place in the Peruvian First Division that season.