The Ecuadorian giants have one foot in the second round after Cristian Penilla, Pedro Velasco and substitute Ismael Blanco were all on target at the Estadio Monumental Isidro Romero Carbo.
Barcelona - unbeaten at home in 10 games in all competitions prior to Thursday's clash - hit the front through Penilla in the 32nd minute.
Christian Suarez drove towards the penalty area and squared the ball for Penilla, who fired beyond Alianza Lima goalkeeper George Forsyth from the edge of the box.
Barcelona doubled their lead against the Peruvian side in the 67th minute, when Velasco beat the keeper with a powerful effort that found the bottom corner.
It got better for the home team in the first minute of injury time thanks to Blanco's header that left Alianza Lima with a mountain to climb ahead of next week's return leg at the Estadio Alejandro Villanueva.
Deportivo Capiata were the only other team to win at home in the first round on Thursday, accounting for Danubio 3-1 in Paraguay.
Francisco Escobar's brace - both penalties - and a goal from Blas Irala gave Deportivo Capiata the edge over their Uruguayan opponents at the Estadio Feliciano Caceres.
Though, Danubio are still in the tie thanks to Matias Zunino, who netted a valuable away goal with eight minutes remaining.
Cesar Vallejo, River Plate, Independiente del Valle and Caracas were all triumphant on the road.
Two first-half goals from Andy Pando gave Peruvian club Cesar Vallejo a 2-1 win over Millonarios in Colombia.
Uruguay's River Plate prevailed 1-0 against Universidad Catolica in Chile.
Venezuelan hosts' Trujillanos slumped to a 1-0 loss to Ecuador's Independiente del Valle, who were helped by Daniel Angulo's 79th-minute strike.
Meanwhile, Dani Cure inspired Caracas of Venezuela to a 1-0 victory at Inti Gas in Peru.
