The move was widely expected after lacklustre performances left the gruff Argentine increasingly embittered.
"When I got here [to Costa Rica] I talked of a project... of building a new national team," the 59-year-old told a news conference.
"What for me is the most difficult thing, an understanding with the locally-based players, has become complicated," added La Volpe, a goalkeeper in Argentina's 1978 World Cup-winning squad who ended his playing career in Mexico.
In 17 matches, however, the "Tricolor" won just four, drew eight and lost five, prompting commentators to slam the coach for failing to turn out a winning team.
Despite the record, La Volpe can be credited for kick-starting the career of Joel Campbell by fielding the young Deportivo Saprissa striker in last month's Copa America in Argentina.
Campbell's performances wowed scouts in Europe and have since earned the 19-year-old a move to Arsenal in England's Premier League.
Mexican media have linked La Volpe with a move to Estudiantes AUG, who play in Mexico's first division where the Argentine spent the greater part of his coaching career.
