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.
Known for his thick mustache and fierce scolding of players, La Volpe signed a $50,000-a-month deal in September 2010, vowing to lead the side to the 2014 World Cup finals in Brazil.
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.
La Volpe, who steered Mexico to the second round at the 2006 World Cup in Germany, leaves behind a cloud of uncertainty over a team that faces Brazil in a friendly on October 7.
Costa Rica, who made the quarter-finals of the regional CONCACAF Gold Cup in the United States in June, have reached the World Cup finals three times, the last in 2006.
