Cocca takes over after Martino's contract expired after Mexico were eliminated in the World Cup group stage last year for the first time since 1978.
"Diego showed his interest in coaching the national team and we are happy to unveil him," said Rodrigo Ares de Parga, Mexico's Director of National Teams.
"He is a hard-working man, he knows Mexican football, he is a great tactician, he has great team management and we believe that starting in March we are going to start getting good results."
The appointment comes a day after Mexican league side Tigres UANL, who Cocca joined just three months ago, announced the termination of the coach's one-year contract after he told the club he had decided to take the Mexico job.
"For me it's not just any job to be the coach of this country that has given me so much, that has opened doors for me and made me grow as a professional and again as a coach," Cocca told reporters.
"To have the chance to help Mexico grow its national team is a privilege and an opportunity that I could not miss."
The former Racing Club boss is known in Mexican football for winning back-to-back league titles with Atlas in 2021 and 2022, ending the club's 70-year title drought.
Mexico are aiming to qualify for the CONCACAF Nations League finals in June, while they will also look to reclaim the Gold Cup, which runs from June 24 - July 16 in the United States, after losing their title to the US in the 2021 final.
The US, Mexico and Canada will co-host the 2026 World Cup.
