Thiago Almada scored the winner with a powerful curled shot from the edge of the box on Friday, putting the defending World Cup champions one point away from a direct spot in the tournament next year.
Argentina lead the round-robin competition with 28 points after 13 matches, and could secure their place with a home draw with Brazil on Tuesday. Coach Lionel Scaloni's team already are 15 points ahead of 7th-placed Bolivia, with only five rounds to the end of the competition.
The last time the two powerhouses clashed, Argentina won 1-0 at the Maracana Stadium in Rio de Janeiro.
Earlier, Ecuador defeated Venezuela 2-1 to move into second place in South American World Cup, edging closer to reaching the tournament once again.
Brazil are third with 21 points, one ahead of Uruguay and Paraguay. Colombia, with 19 points, own sixth position.
The top six teams will secure direct berths to the tournament in the United States, Mexico and Canada. The seventh-place team among the 10 of the region will still have a chance to qualify through an international playoff.
Argentina had to accommodate key absences beyond Messi. Lautaro Martínez, Lisandro Martínez, and Rodrigo De Paul did not play in Montevideo. Scaloni gave a place to Giuliano Simeone in the starting lineup, with Julián Álvarez and Thiago Almada up front.
Uruguay, playing in a full Centenario Stadium against their arch-rivals, took the initiative and had the clearest chances in the first half. But Argentina managed to hold their ground and produced better opportunities after half-time.
Almada's goal in the 68th minute, after Argentina had wasted several other opportunities in front of goalkeeper Sergio Rochet, still didn't push Uruguay forward as Marcelo Bielsa's team finished the match with only four shots on goal.
"I was a little anxious, very willing to play and to show why I was making the squad," said Almada, a world champion with Argentina in 2022. "Now we have to rest and wait for this match against Brazil, they will rest one more day than us."
Argentina coach Scaloni celebrated his team's performance despite so many absences.
"We made a complete match, we absorbed the pressure. When we had to play, we did it. And when we had to defend, we did it too," Scaloni said in a press conference. "I am happy not only for our win, but also for how the team behaves."
Ecuador, meanwhile, started the competition with a three-point deficit due to fielding Byron Castillo, who was actually Colombian and ineligible to play in qualifying for the 2022 World Cup.
Now, they are close to qualifying.
Enner Valencia, Ecuador's all-time top goalscorer, added another two to his tally. He broke through the Venezuelan defence in the 39th minute and scored with a powerful right-footed shot from an acute angle.
In the first minute of the second half, Valencia added the second after a run on the left flank ended in a classy poach over goalkeeper Rafael Romo.
Valencia could have had a third from the spot in the 69th but missed.
Jhonder Cádiz scored for Venezuela in injury time.
Venezuela have not won for nine straight games but remain in eighth place and still have a chance of qualifying for the first time in history.
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