It was four-times South American champions Estudiantes' worst defeat outside Argentina in the competition and Alex Sabella's first loss as their coach in the Libertadores Cup.
Alianza got off to the worst possible start after kicking off and playing the ball back into the centre of defence.
They conceded possession to Jose Sosa, who ran into the box, rounded goalkeeper Salomon Libman and netted the fastest goal in South America's top club competition.
However, Alianza set about playing a fast, quick-passing game that soon had Estudiantes on the back foot.
Aguirre leveled the score in the 17th minute, racing into the box and blasting over keeper Agustin Orion.
The man of the match burst through again just past the half hour to control a high ball out of his own half, fend off two defenders and place a low shot inside the far post.
His third goal in the 74th minute was a low shot inside the far post after he twice beat tall centre-back Leandro Desabato.
Central striker Juan Carlos Fernandez volleyed a brilliant fourth seven minutes from the end to chants of "Ole" from the home crowd.
Peru has not been a happy hunting ground for Estudiantes over the years with three defeats and a draw.
"I thought the worst," Alianza's Argentine coach Gustavo Costas told Fox Sports of the early goal. "But the team showed their maturity."
"We are very committed, we want to change Alianza history (in the Libertadores Cup)," said Fernandez, whose club's best performance was reaching the semi-finals in 1976 and 1978.
Estudiantes captain Juan Sebastian Veron said: "It wasn't a good night, above all defensively, which is normally one of our strengths."
There was also a quick strike in Chile where Uruguayan Juan Manuel Olivera converted a second-minute penalty for Universidad de Chile against Caracas FC of Venezuela.
But there were no more goals and the home side won 1-0 in Group Eight.
