Scoring all the goals in the 3-0 win in Buenos Aires brought Messi's total for Argentina to 79 in 153 games.
Brazil legend Pele finished his career on 77 international goals in 92 games.
Messi's 14th-minute goal from outside the box put him level with Pele and he broke the record in the second half.
The 34-year-old dribbled past a defender for his second goal before completing a hat-trick from a rebound two minutes before full-time.
After the match, Argentina captain Messi and his team-mates celebrated with the Copa America trophy in front of 20,000 fans at the Monumental Stadium.
He led the side to that title, his first major trophy in international football, in July, beating Brazil in the final.
"I dreamed about this day a lot and thanks to God it arrived," Messi wrote on Instagram alongside a picture of the team with the trophy after the match.
"I don't have words to thank you for all the love I've received. What a beautiful night, I enjoyed it so much - unforgettable."
Messi's 29-year-old Paris St Germain team-mate Neymar has scored 69 goals in 113 appearances for Brazil.
Argentina are second to Brazil in the South American World Cup qualifying table after winning five and drawing three games.
Marta holds the record for most goal scored by a South American player, having notched 109 for Brazil's women's team.
