In a match that saw both teams reduced to 10 men in the first half on Wednesday, Alianza Lima came from behind after Koichi Aparicio turned the ball past his own goalkeeper in the 17th minute.
Uruguayan midfielder Costa restored parity after 35 minutes, surging between two defenders before firing the ball beyond Fernando Martinuzzi.
Alianza Lima midfielder Christian Cuevas saw red for a second bookable offence for simulation three minutes later.
Los Caimanes' numerical advantage did not last long after Daniel Hidalgo was sent off in the 43rd minute.
Costa netted his second goal two minutes after the break as Alianza Lima hit the front and Julio Landauri completed the scoring 10 minutes from time with a powerful strike.
Alianza Lima have collected 27 points from 11 matches and are one point clear of Sporting Cristal.
Carlos Lobaton struck in the 90th minute as Sporting Cristal overcame Sport Huancayo 2-1 on Tuesday.
In other results, quick-fire goals from Victor Balta and Omar Fernandez saw Melgar to a 2-1 victory at third-bottom Cesar Vallejo.
Carlos Preciado scored twice in the final five minutes as Leon de Huanuco accounted for fourth-placed Union Comercio 2-0 to end a run of three consecutive defeats.
Juan Aurich were 1-0 winners against Real Garcilaso, Universitario were held to a 1-1 draw at second-bottom San Simon, while Cienciano and UTC Cajamarca drew by the same scoreline.
Universidad San Martin's match against Inti Gas on Thursday was postponed.
