The tournament hosts created the match's first chance after five minutes, but Vargas's shot was blocked by Christian Cueva.
Peru came closest to scoring in the first half, though, when Jefferson Farfan's header hit the post after nine minutes.
Carlos Lobaton was next to come close to giving Peru the lead, but he hit the side-netting.
Chile ramped it up a notch following Zambrano's red card, and Jorge Valdivia curled the ball wide of the target on the half-hour mark.
With nine minutes of the first half remaining, Vargas saw his shot blocked on the line by Luis Advincula.
Chile took the lead in the 43rd minute when Vargas tapped in from a rebound after Alexis Sanchez's cross hit the post.