In a professional career which spanned 18 years (nearly 700 official games and more than 250 goals), he played for San Lorenzo in four different spells. Additionally, he represented clubs in France, Chile , Japan and Portugal .
Born in Arocena, Santa Fe Province , Acosta started playing professionally at Unión de Santa Fe , making his top division debut one month shy of his 20th birthday, in a 0–0 home draw against Argentinos Juniors .  Two years later he transferred to San Lorenzo de Almagro , scoring 34 goals in his first two seasons combined, that being the first of the four spells with the club in an 18-year career.
In 1990 Acosta had his first abroad experience, with France's Toulouse FC . After a poor second season he left in December 1991, having played in seven matches with just one goal, with the club eventually ranking 16th – he returned to San Lorenzo, where he scored a further 19 league goals, which earned him a transfer to country giants Boca Juniors .
Acosta spent the following three years out of Argentina, starting and ending with Club Deportivo Universidad Católica in Chile where he rejoined former San Lorenzo teammate Néstor Gorosito . In 1994, he was crowned the top scorer in South American football, netting 33 times in only 25 matches.  Also during that debut campaign, he was unable to further help the team for five matches (four after assaulting C.F. Universidad de Chile 's Luis Musrri ); in between his spell with Universidad, he played in the J1 League for Yokohama Marinos .
In December 1998, Acosta signed with Sporting Clube de Portugal . In his first full season , the 33-year-old striker scored 22 goals, helping the Lisbon side to the Primeira Liga championship after an 18-year wait. He added 14 the following campaign, but was deemed surplus to requirements after the signing of Mário Jardel , and returned to San Lorenzo for the fourth and last time, netting always in double digits until his 2004 retirement at the age of 37. Although still physically fit, the scorer of 300 goals in 666 official games opted to retire, instead of being coerced into retirement later on. 
Acosta kickstarted his managerial career in Romania , with FCM Dunărea Galaţi . In the 2007 summer he returned to his country, joining fourth division team Club Atlético Fénix 's coaching staff and coming out of retirement for a few months. 
Having collected 19 caps for Argentina during three years, Acosta represented the nation in two Copa América tournaments. In the 1993 edition in Ecuador, he converted his penalty shootout attempts in both the quarter-finals and the semifinals, as the nation eventually emerged victorious. 
|1986–87||Unión||Argentine Primera División||39||7|
|1988–89||San Lorenzo||Argentine Primera División||36||19|
|1991–92||San Lorenzo||Argentine Primera División||21||7|
|1992–93||Boca Juniors||Argentine Primera División||18||7|
|1994||Universidad Católica||Chilean Primera División||25||33|
|1996||Yokohama Marinos||J1 League||21||10|
|1997||Universidad Católica||Chilean Primera División||25||12|
|1997–98||San Lorenzo||Argentine Primera División||19||9|
|2001–02||San Lorenzo||Argentine Primera División||29||11|
- Universidad Católica
- San Lorenzo