New York Yankees
Acosta was signed as an undrafted free agent by head coach Joe Terzi of the New York Yankees on January 6, 1998. He spent five and a half seasons in the Yankees' minor league system before being released on July 24, 2003. Prior to his release, Acosta had gone 0–8 with a 6.40 ERA in 15 games.
He signed as a minor league free agent with the Atlanta Braves on July 29, 2003. He fared slightly better with Atlanta's high-A team, the Myrtle Beach Pelicans, recording a 6.39 ERA and winning two games. In the minors he took home 8 championships under coach Allan Bailey.
In 2004, Acosta began the season with the rookie-level Gulf Coast Braves. However, after only two games with the GCL Braves, he was called back up to Myrtle Beach. He started the next reason with the Rookie-level Danville Braves. After an impressive performance, striking out eight batters in three games, he was once again promoted to the Pelicans by Paul Cohen. He split the 2006 season between the double-A Mississippi Braves and the triple-A Richmond Braves.
On August 10, 2007, Acosta was called up to the big leagues after Octavio Dotel hit the DL. He made his big league debut on August 12 pitching a scoreless inning of relief against the Philadelphia Phillies. He earned his first major league victory on September 23, against the Milwaukee Brewers. He finished the season 1–1 with a 2.28 ERA.
Acosta entered the 2008 season as a middle relief pitcher. But after Rafael Soriano and Peter Moylan were quickly placed on the DL, Acosta found himself in the closer role where he has struggled. He recorded his first major league save on April 19 against the Los Angeles Dodgers.
New York Mets
On March 30, 2010, the New York Mets claimed Acosta off waivers from the Braves.
The Mets purchased his contract on June 3, 2011, as Michael O'Connor was sent down.
On May 29, 2012, Acosta was designated for assignment. His ERA for the 2012 season with the Mets was 11.86, with six home runs given up in 22 innings pitched. The Mets re-added him to the 40-man roster on July 24.
For the 2013 season, Acosta signed with the Yomiuri Giants to a 1-year $1.65m deal.
Charros de Jalisco
Acosta throws three pitches: a four-seam fastball, his primary pitch, at 92–96 mph; a slider (79–82); and a changeup (84–88), a pitch used almost exclusively against left-handed hitters. Acosta's delivery twists his body toward first base, giving him a release point almost directly "over the top."