Messi, 36, opened the scoring after 32 minutes and added a second with a spectacular finish moments later.
He now has 31 goals in Conmebol World Cup qualifiers, surpassing Luis Suarez 29 for Uruguay.
"This team is incredible. Every time they play they are very close to being the best in history," Messi said.
"On a game level I think we have grown. After winning the World Cup we are confident, very loose, more united and firm. Hopefully we can continue to grow."
Forward Messi missed by Houston Dynamo last month because of injury and had not started a match for his club since 21 September.
He came on as a substitute in Argentina's 1-0 win over Paraguay on Thursday and started in Lima on Tuesday.
Argentina head coach Lionel Scaloni said: "The team has understood Messi for a long time. It benefits him and he feels comfortable.
"I hope he plays as much as he can because everyone is happy to see him on the pitch."
A tearful Neymar was carried off on a stretcher in Brazil's 2-0 defeat by Uruguay after suffering what appeared to be a serious knee injury.
The 31-year-old forward, who Joined Al-Halal from Paris St-Germain in August, landed awkwardly after challenging Nicolas de la Cruz for the ball in the first half in Montevideo.
"Let's hope it's nothing serious," Brazil captain Casemiro told Globo television.
"He's an important player for us; we're very fond of him. He's been suffering a lot from injuries and when he starts to pick up the pace he gets injured again."
Liverpool forward Darwin Nunez scored Uruguay's opener with a header and set up De la Cruz for their second as Uruguay beat Brazil for the first time since 2001.
In Tuesday's other qualifiers, Venezuela beat Chile 3-0, Paraguay defeated Bolivia 1-0 and Ecuador drew 0-0 with Colombia.
Argentina, who remain top of the Conmebol standings, play second-placed Uruguay and third-placed Brazil in November.
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