Argentina had multiple chances to score throughout the 90 minutes – with Messi hitting the post from point-blank range and curling a first-half shot just wide of the post, ESPN reported.
The result leaves the South American giant in sixth place in the South American table, outside of the top four spots that guarantee automatic qualification and the fifth-place spot that forces a playoff against New Zealand for a berth in Russia.
Messi, who would likely be playing in his last World Cup should the team qualify, has yet to deliver a major trophy for Argentina, but his manager, Jorge Sampaoli, is still confident Argentina will find a way.
"Our situation is not very comfortable," Sampaoli said. "But it depends on us."
Argentina has scored just one goal in its last three qualifying matches under new coach Sampaoli after he took over for Edgardo Bauza before the summer, drawing all three under the former Sevilla manager.
"I remain confident we'll be in the World Cup," said an upbeat Sampaoli after the match.
"It would really be unfair if a group that put it all out there – like they did – does not advance.
"Throughout the game we had a marked advantage against them, but we could not break through."
Argentina is two places and two points behind Colombia, which is fourth and holds the last direct qualifying spot. Sampaoli's team takes on Ecuador in Quito on Tuesday in its final qualifier.
"We are in this situation," Sampaoli told a news conference. "If we confront it with the conviction that we had against Peru, we will win at Ecuador.
"In the changing room, there was a feeling of enthusiasm when we found out that if we win in Quito, we will qualify.
”I'm going to take advantage of that chance and so are the players.
"Our group of players competed to the standard required by me and that allows us to be optimistic heading into our next game."
Sampaoli praised the commitment and overall play of Messi. The Argentina captain left the changing room with his head down as the Albiceleste players continued their media silence. The players, believing they were not being treated fairly by the local press, have not spoken to reporters since November 15, 2016.
You can't ask anything more from him,” Sampaoli said.
"Leo played the game with a lot of intensity, he applied pressure, had chances, he never stopped for a second to find the ball, he created goal-scoring opportunities in impossible angles and we saw a very intense Messi which is what we need."
Argentina, runner-up to Germany three years ago in Brazil, plays its final match on Tuesday against Ecuador in the thin air of Quito in the Andes and can guarantee at least a spot in the playoff with a win.
However, Argentina has won just once away to Ecuador since 1960 and would become the first country not to qualify for the World Cup after reaching the final since Czechoslovakia in 1962 and 1966.