Alianza leapfrogged Union Comercio in the standings on Sunday after the latter were held to a 2-2 draw at home by Real Garcilaso.
Costa's first goal was a powerful header four minutes before the break, while he struck his second in the 65th minute, thanks to his persistence.
The Uruguayan central midfielder got clear on the right, burst into the box and ducked inside an opponent before his first shot was saved by Leon goalkeeper Jaime Rivera, but Costa followed up to half-volley the ball into the net.
The win saw Alianza move to 15 points from seven matches, two ahead of Union, who gave up a two-goal lead against lowly Garcilaso.
Cristian Bogado struck both goals for Union in the first half - the first with a header, the second from the spot - but the visitors pulled a goal back two minutes before half-time when Ramon Rodriguez volleyed a long ball into the net from the edge of the box.
Rodriguez then completed his brace with a header with 10 minutes remaining to take Garcilaso level, and despite having Marcos Delgado sent off in the 85th minute, the away side held on for a draw.
In the other game on Sunday, Sport Huancayo defeated Cesar Vallejo 3-1.
Leaders Melgar came from behind at home on Saturday to draw 1-1 with fourth-placed Universitario with Omar Hernandez striking the equaliser in the 92nd minute.
Miguel Torres had given Universitario the lead in the 21st minute but they missed the chance to finish the weekend in third after Hernandez's late heroics, while Melgar ended the match with 10 men after Diego Pizarro was sent off just after the equaliser.
Los Caimanes and Sporting Cristal had the biggest wins of the weekend, crushing Universidad San Martin and Juan Aurich 4-1 respectively.
Cienciano trumped San Simon 2-0 and Inti Gas won 1-0 at UTC Cajamarca.