The six-time Ballon d'Or winner scored a hat-trick in Thursday's 3-0 World Cup qualifying win over Bolivia to overtake Brazil great Pele as CONMEBOL's top-scoring player.
Messi's 79 goals for Argentina have come in 153 appearances, compared to 77 in 92 caps for Pele, who had held the record since retiring from international duty in 1971.
It is the latest on a long list of records to have been broken by the ex-Barcelona star for club and country, and it comes two months after helping Argentina to win the Copa America.
La Albiceleste paraded the trophy – their first in 28 years – around the pitch at the Monumental at full-time and an emotional Messi broke down in tears.
Scaloni embraced Messi on the field and later hailed the Paris Saint-Germain forward for continuing to prove decisive for his country following his latest treble.
"I am proud to have him as a player. Messi is leaving a legacy to all the soccer players of the national team," Scaloni said at his post-match news conference.
"Messi was always a player who made a difference. Before and now. Winning magnified everything, and if before he was already an idol of the people, now it is much more.
"Seeing them excited fills us with joy, but it lasts [only] until this international window is over, because afterwards we have to continue working."
Messi's hat-trick was his seventh for Argentina but his first on home soil and it helped extend his country's unbeaten run to 22 matches in a run stretching back to July 2019.
He is the only Argentina player to have scored multiple hat-tricks in World Cup qualifying.
Scaloni's men are second in the 10-team qualifying group, six points behind Brazil, whom they still have to face after last week's high-profile clash was controversially suspended.
Argentina also defeated Venezuela 3-1 in the same international window and Scaloni is pleased with what he has seen over the past week.
"I think we have a line of work from day one and we have grown," he said. "We've improved and corrected things even when we've won.
"This team has sacrifice, humility and an ability to unbalance other players in one-on-one situations."