Costa scored the first after 21 minutes and then laid on the second for Renato Augusto 12 minutes into the second half. Filipe Luis grabbed a third in 76 minutes after the keeper parried Costa's ferocious drive from the right wing.
"It was very important to win after the draw in Argentina and we did well," said Luis, who scored his first international goal. "The team is growing and getting better every game and we are finding a style of play that we want."
Brazil, who made three changes from the side that were fortunate to draw 1-1 with Argentina in Buenos Aires on Nov. 13, made a shaky start to the game.
Peru's Paolo Guerrero should have done better when he only had the goalkeeper to beat after just three minutes but goalkeeper Alisson got down well to smother his shot.
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Brazil, however, opened the scoring after 21 minutes when Douglas Costa got on the end of a cross from Willian to bundle the ball home from close range.
The 25-year old was unlucky not to get a second after 39 minutes when his swerving cross bounced off the post and away to safety.
Peru's Christian Cueva had an overhead kick well saved by Alisson in the 53rd minute before Willian watched Diego Penny parry a 20-yard drive to safety a few moments later.
Brazil doubled their lead after 57 minutes when Costa laid the ball into the path of Renato Augusto, who stroked the ball home from 20 yards out.
Neymar, who was regularly fouled by the Peruvians, was unlucky to see his goal cruelly chalked off for offside midway through the second half.
Luis made it three with 14 minutes left when he was on hand to fire home the rebound from Costa's parried shot.
The result lifts Brazil into third in the 10-team table, one of three sides on seven points.
(Writing by Andrew Downie; Editing by Greg Stutchbury)
