Victory on Tuesday night (Wednesday AEDT) 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. The top six teams will automatically qualify.
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 (Wednesday AEDT), 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 minute.
Uruguay have 20 points, two ahead of five-time World Cup winners Brazil after their second straight draw.
After a tight first half, Real Madrid midfielder Valverde opened the scoring in the 55th minute when he curled a long-range low shot into the far corner.
Gerson brought the hosts level seven minutes later, the Flamengo midfielder scoring his first international goal with a terrific volley after a poor clearance by the Uruguayan defence.
Gerson scored the equaliser against Uruguay, but said he would swap his goal for a Brazil victory. (AP PHOTO)
"I'd actually swap my goal for the win," Gerson told Brazil's Globo TV.Â
"We knew it was a tough game and we wanted to win. We conceded the goal and got the equaliser.
"We'll just have to keep working at our clubs and next year when there's another international break we can go for it. We just have to keep working and trust in our work."
Brazil pushed for a winner, but late efforts from Luis Enrique and Raphinha went wide as boos from home fans at the Arena Fonte Nove rang out at the final whistle.
"I think the booing is more about the result, because in my opinion we gave it all," Raphinha said.
"I'm proud of those who played and even those on the bench. We did everything we could to get the result."
Earlier, third-placed Ecuador won 1-0 at fourth-placed Colombia with a goal by Enner Valencia in the seventh minute.
View this post on Instagram A post shared by SBS Sport (@sbssportau)
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 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 to beat Venezuela 4-2.
The next round of South American World Cup qualifying will be played in March.
With AAP.
30°C