Argentina, who had already qualified for the quarter-finals after wins over Canada and Chile, finished top of the group, while Peru were eliminated.
Captain Lionel Messi was rested for the final group game, one of nine changes to Argentina's starting line-up - but the Albiceleste were still largely untroubled.
Martinez finished off a flowing move in the 47th minute, coolly chipping the ball over Peru keeper Pedro Gallese to put his side ahead.
He repeated the trick four minutes from time, again chipping Gallese to bring up his second goal after some sloppy defending by Peru.
Argentina also missed a 72nd-minute penalty when Leandro Paredes smashed his effort off the post.
They will be joined in the knockout stage by Canada, who qualified for the last eight after holding Chile to a 0-0 draw in Group A's other match on Saturday.
Chile finished third to exit in the group stage for the first time since 2004.
Needing to win and have other results go their way to qualify, Chile's task was made even harder when Gabriel Suazo was shown a second yellow card in the 27th minute for bringing down Richie Laryea.
In a game in which chances were few and far between, Chile goalkeeper Gabriel Arias did well to keep out shots from Canada's Stephen Eustaquio and Tajon Buchanan, while Canada stopper Maxime Crepeau made a smart save to deny an awkward deflected effort from Alexis Sanchez.
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