Tuesday night's victory nearly secures Lionel Messi's team a spot in the 2026 finals in the United States, Mexico and Canada.
Messi crossed the ball into the penalty area in the 55th minute to give Martinez one of the few clear opportunities of the match.
Argentina lead the standings with 25 points from 12 matches. The loss left Peru (seven) in the 10th and last position in the round-robin competition.
Until Martinez scored, Argentina's best chance was in the 21st minute, when striker Julian Alvarez hit the Peruvian goalkeeper's right post. Messi once again had a modest performance, as he did in his team's 2-1 defeat at Paraguay.
Also on Tuesday, Uruguay moved up to second place after drawing 1-1 in Brazil.
Federico Valverde opened the scoring for the visitors on 55 minutes before Gerson levelled with a brilliant strike in the 62nd.
Uruguay have 20 points, two ahead of five-time World Cup winners Brazil after their second straight draw.
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Earlier, third-placed Ecuador won 1-0 at fourth-placed Colombia with a goal by Enner Valencia in the seventh minute.
He dribbled through three players before scoring. Ecuador lost defender Piero Hincapie to a red card in the 34th minute but held on for the win, leaving both nations on 19 points.
Bolivia and Paraguay drew 2-2 in a match between teams that could end up fighting for the seventh position, which secures a berth in an international play-off.
Chile, one of the worst performing teams in the region this year, showed some fighting spirit and beat Venezuela 4-2.
The next round of South American World Cup qualifying will be played in March.
With AAP.
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