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Di Maria started for just the second time in the Copa, and he justified his selection by scoring the opener midway through the first half.
Brazil piled on the pressure in an exciting second half, but even with five strikers on the field, they could not get an equaliser against an Argentine defence protected by the outstanding De Paul.
"I can't cry yet,'' Di Maria said after the match. "We dreamed of getting this, and we fought. Many people criticized us and asked us not to come back [to the national team].''
The match itself was a disappointing one, with Argentina the better side in a cagey first half that featured 21 fouls.
"First, we have to congratulate our opponents, especially for the first half, when they neutralised us," Brazilian defender Thiago Silva said.
"In the second half, there was no contest -- only one team tried to play football; the other just wasted time, as we knew they would. It's not an excuse; we didn't do what we had to, principally in the first half."
The Argentine players surrounded their captain at the final whistle.
Goalkeeper Emiliano Martinez celebrated what he called a Maracanazo, a remarkable win at the famous Rio stadium.
"I'm speechless," he said. "I knew my dream would come true, and where better than the Maracanazo and giving the title to the best in the world and fulfilling his dream."
Messi finished as the tournament's joint top goal scorer with four goals and was elected joint best player along with Neymar. But Messi was quiet throughout Saturday's contest and uncharacteristically missed a golden opportunity to wrap the game up with two minutes remaining.
When the final whistle went, Argentina TV declared, "Argentina Champions, Lionel Messi Champion!"
"This is a very big title," Argentine coach Lionel Scaloni said. "I hope that Argentines can enjoy it. The fans love the team unconditionally, and I think they identify with this side that never drops its guard."
The victory was Argentina's 15th Copa America triumph, and it means they draw level with Uruguay as the all-time leading winners.
Argentina's win extended their sequence of undefeated matches to 20 under Lionel Scaloni and handed Brazil their first competitive defeat since they lost to Belgium in the quarterfinals of the 2018 World Cup.