Acosta began playing football at FC Arocena in his hometown. In 1982 he moved to Unión de Santa Fe. His debut in the first team he gave on July 13, 1986 against Argentinos Juniors (0-0 ). His first goal in La Liga scoring striker on 24 August of the same year with a 1-2 defeat against CA Independiente. After he played from 1988 to 1990 in San Lorenzo de Almagro, he moved to season 1990/91 to France for Toulouse FC. In his first season in the premiere division he was with six goals together with his compatriot Alberto Márcico internal top scorer of the club. Despite a 17th place in the final table, we stayed in the league, as three teams had relegated. 1991/92 lost Acosta his place and even went during the current season returned to Argentina to San Lorenzo de Almagro. 1993 obliged him Boca Juniors.
After two seasons there, he moved to Chile to based in Santiago club CD Universidad Católica. There he was successful with 33 goals in only 25 games in the first year (1994 ), which he finished as top scorer. He was also able to celebrate with his teammates winning the Copa Interamericana. Acosta was elected Chile's footballer of the year 1994. In 1995, he was top scorer of the Copa Chile (ten goals), which he won with his club. In addition, Universidad Católica was División in both years runner-up in the Chilean Primera. In 1996, Acosta for Japanese club Yokohama F. Marinos, where traditionally play many Argentines. In 1997 the return to CD Universidad Católica. He was prompt with his club Chilean champion. After CD Universidad Católica and Colo -Colo were equal on points in the final table was Universidad Católica in the playoffs successful ( 0:1 and 3:0). Alberto Acosta scored the 1-0 in the second leg. In 1998 he signed for the third time at CA San Lorenzo de Almagro.
In December 1998 he went to Portugal to Sporting Lisbon. In the first year there were disagreements within the team and Acosta's goal tally amounted to only three goals in 13 games. At this time, Acosta also suffered from back problems. His coach Giuseppe Materazzi called for in a letter to the club's management even dismissal Acosta because he was " a disturbing element in the group " and that the team split. However, the club management not complied with this request, but released later Materazzi itself Then it ran better for Acosta and Sporting. The club was 1999/2000 champion, Acosta contributed to 22 goals for, which meant second place in the scoring charts. 2000/ 01 he played for the first time in the Champions League. However, Sporting separated already from in the first round and in six games Acosta managed a goal. In the league Acosta scored 14 goals again and returned to his homeland, where he is a fourth time San Lorenzo de Almagro joined afterwards. 2004, with 37 years, he decided to end his career.
In 2006, he began his coaching tenure at the Boca Juniors. He then coached the Romanian club Dunărea Galaţi (2006 to 2007). In 2008, he made his comeback as a player with Club Atlético Fénix, who competes in the Primeira C, the fourth highest football league in Argentina. He also was coach of the team.
In 2011 it was announced that Acosta has been diagnosed with thyroid cancer.