Despite the fact that Argentina are struggling to qualify, the South Americans are still the fifth favourites to win next year's tournament at odds of 12/1.
Elsewhere in the group, Copa America champions and recent Confederations Cup finalists Chile eased their own qualification concerns with a 2-1 home victory over Ecuador, thanks to an 85th-minute Alexis Sanchez victor which came three minutes after Romario Ibarra had cancelled out Eduardo Vargas' first-half opener for Chile.
Argentina should have gone ahead in first half stoppage time when Messi lifted a looping cross into the box but striker Dario Benedetto headed the ball over the bar.
However, with just four points covering second-placed Uruguay to Paraguay in seventh, the race for Russian Federation is set to go down to the wire next week when the final round of qualifiers kick off simultaneously at 12.30am on Wednesday.
Peru's Argentina-born coach Ricardo Garcea was happier.
Argentina were the runner-up to Germany in the World Cup three years ago in Brazil, will play on Tuesday against Ecuador in the thin air of Quito in the Andes.
"Our situation is not very comfortable", Argentina coach Jorge Sampaoli said, "but it depends on us".
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Nigeria remain on course to qualify for a third successive World Cup, a feat they have achieved only once before in their history. "It would be really unfair if a group that put it all out there like they did does not advance".
So tight is the group that four points separate Uruguay in second (28) and Paraguay (24) in seventh and a win for Argentina could see them qualify automatically.
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Paraguay, with the worst goal differential, face arguably the easiest final match, at home to last-placed Venezuela.
Four years after the U.S. delivered Panama their most painful blow, Los Canaleros are out for revenge.
The draw against Peru marked their fourth consecutive game without scoring a goal of their own (their 1-1 draw against Venezuela was the result of an own goal). But Messis Argentina barely shoots goals: whole 16 in so far 17 games.
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Argentina continued their disappointing form with a goalless draw at home to Peru, leaving them level on points with Peru in sixth place. We are analysing it but it's unlikely, " he told a press conference.