New York Yankees Acosta was signed as an undrafted free agent by the
New York Yankees on January 6, . He spent five and a half seasons in the Yankees' minor league system before being released on July 24, . Prior to his release, Acosta had gone 0–8 with a 6.40
ERA in 15 games.
Atlanta Braves He signed as a minor league free agent 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 Teddy Beyda. In , 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. He split the season between the double-A
Mississippi Braves and the triple-A
Richmond Braves. Acosta pitched for
Panama in the
2006 World Baseball Classic. He had an 0–1 record during the event. On October 12, 2006, the Braves purchased Acosta's contract and added him to their 40-man roster. On August 10, , 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.
Yomiuri Giants On December 13, 2012, Acosta signed a one-year, $1.65 million contract with the
Yomiuri Giants of
Nippon Professional Baseball. Acosta made 14 appearances for Yomiuri in 2013, compiling a 1-0 record and 5.54 ERA with 12 strikeouts over 13 innings of work.
Diablos Rojos del Mexico On March 29, 2014, Acosta signed with the
Diablos Rojos del México of the
Mexican League. In 52 appearances out of the
bullpen, he logged a 6-4 record and 3.96 ERA with 59 strikeouts and 20
saves over innings pitched. During the offseason, Acosta pitched for the
Charros de Jalisco in the
Mexican Pacific League, a winter league. Acosta made 2 scoreless appearances for México during the 2015 season, striking out 3 batters across innings pitched. Acosta pitched in 49 contests for the Diablos in 2016, compiling a 6-5 record and 4.40 ERA with 50 strikeouts and 14 saves across innings pitched. He was released by the Diablos on September 23, 2016.
Sultanes de Monterrey On February 20, 2018, Acosta signed with the
Sultanes de Monterrey of the
Mexican League. In 40 appearances for Monterrey, he posted a 3.94 ERA with 34 strikeouts across innings of work. Acosta was released by the Sultanes on August 10.
Diablos Rojos del México (second stint) On August 16, 2018, Acosta signed with the
Diablos Rojos del México of the
Mexican League. In 7 appearances for the team, he posted a 2-0 record and 1.04 ERA with 6 strikeouts and 5 saves across innings pitched. Acosta became a free agent following the season.
Bravos de León On March 12, 2019, Acosta signed with the
Bravos de León of the
Mexican League. In 43 relief appearances for León, he registered a 3-3 record and 3.86 ERA with 52 strikeouts and 17 saves across innings pitched. Acosta did not play in a game in 2020 due to the cancellation of the Mexican League season because of the
COVID-19 pandemic. He was released by the Bravos on March 9, 2021. ==Pitching style==