Angel Di Maria was the matchwinner at Maracana in Rio as Messi finally won something with his country in what was his 10th major tournament.
Messi fell to the ground in joy at full-time, while Neymar was in tears after he was unable to inspire Brazil to victory.
After helping Argentina end their 28-year wait since they last won the Copa America, Messi was named player of the tournament after his four goals in the competition.
The 34-year-old had previously been a beaten finalist on four separate occasions with Argentina – three times in the Copa America and once in the World Cup.
Di Maria scored the winner midway through in the first half with a superb lob over Manchester City goalkeeper Ederson following a perfect pass from Rodrigo De Paul.
Brazil were poor throughout the match and struggled to test Emiliano Martinez at the other end.
Messi will now face questions about his immediate future, given he is currently a free agent after his contract at Barcelona expired two weeks ago.
There is also the prospect of Messi going for what would likely be his last chance of winning the World Cup in 2022, should he lead his country out in Qatar at the age of 35.