The victory puts the Copa America winners, who beat Brazil 2-0 last week, level on six points with Uruguay and Ecuador at the top of the 10-nation group standings.
Chile took an early lead when a through ball from Mauricio Isla found Sanchez in the box and he slotted past goalkeeper Pedro Gallese.
Peru equalised in the 10th minute through Jefferson Farfan but were reduced to 10 men in the 23rd minute after Christian Cueva was sent off for a kick at Jorge Valdivia.
They still managed to grab the lead when Farfan converted a 36th-minute penalty.
Chile pulled level four minutes before halftime when striker Vargas scored from Valdivia's pass and went ahead with Sanchez's second goal in the 44th.
Vargas scored his second four minutes into the second half after running onto a pass from Sanchez and wrong footing Gallese.
A double save by Gallese from Arturo Vidal then Valdivia prevented Chile from scoring a fifth after an hour and Paolo Guerrero struck Peru's third in added time.
(Writing by Rex Gowar; Editing by Greg Stutchbury)
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