Acosta was taken on January 6, 1998 by the New York Yankees under contract. At this time he had the status of a free agent, because he was never selected in the MLB Draft. He then spent more than five years in the minor league teams in the Yankees before he was released in July 2003.
On 29 July 2003 Acosta was then taken by the Atlanta Braves under contract with their minor league teams Myrtle Beach Pelicans, Gulf Coast Braves, Danville Braves, Mississippi Braves and the Richmond Braves, he was subsequently active.
Due to a violation of Octavio Dotel Acosta was appointed to the Major League in 2007. In his first appearance on 12 August 2007 he was allowed to pitch an inning against the Philadelphia Phillies. His first win he succeeded on 23 September against the Milwaukee Brewers.
As of the 2008 season, Acosta was regularly as relief pitcher in the Braves squad. His first save he succeeded on 19 April 2008 against the Los Angeles Dodgers.
On April 30, 2010 Acosta joined the New York Mets, where he was deployed in 2010 41 times in the season.
During the World Baseball Classic in 2006 Acosta pitched for the national team of Panama. There he posted a win- loss of 0-1 and finished with his team in 14th place