Because Brazil had been playing earlier that day, Argentina knew they had to win with a 4 goal difference, no small feat in a match for a place in the final (there was no semi-final, just a second round).
Jorge Videla, the Argentinian dictator visited the Peruvians in their dressing room. He brought his imperial bodyguards and held a speech about Latin American friendship. Some say Henry Kissinger, the Nobel Peace Prize laureate who had Cambodia bombed illegally, was with him.
Peruvian player Rodolfo Manzo said after the match that he was bought, later he retracted his statement and then he got a job with Velz Sarsfield, an Argentinian side.
And then there were economical reasons for Peru to lose. See this
The conspiracy theorists found more facts:
The Peruvian goalie, Quiroga, was born in Argentina
Peru had two substitutes playing instead of regular players who were not injured
Player Roberto Rojas died in a car crash and the trainer, Marcos Calderon, died in a plane crash. A bomb exploded in a stadium were Quiroga was, the man who had told an Argentinian newspaper that some players had been bought.
Videla told captain Lacoste (the Argentinian World Cup committee president) to organize a fix. First Peruvian officials were given money, then the players were accustomed. At least three of them acknowledged having received money (20.000 dollars).
A journalist, Carlos Ares, found out about the bribes and went to see captain Lacoste who told him to shut up and leave the country or else.
There was a thing called Operation Condor, not a William Forsyth novel, but a political prisoner exchange program between dictatorships. You torture ours and we will torture yours. Then Peruvian president Francisco Bermudez had sent 13 prisoners to Videla and promised things for the favour. One of those things, according to Genaro Ledesma a former senator and one of the 13, was the promise to lose the match.
Ledesma said so during a trial in 2012 against Bermudez, both very old men by then.