Maria Sharapova, a second seed, smacked six aces and connected on 69 percent of her first serves to win in two hours and 24 minutes.
She denied Peer on 10-of-13 break-point chances while taking half the 10 she earned for herself.
Three-time Grand Slam singles champion Sharapova will next face the winner of a later match between American Sloane Stephens and Italy's Sara Errani.
World No. 2 Sharapova lost to top seed and World No. 1 Victoria Azarenka of Belarus in the Indian Wells final last Sunday in her most recent prior match.