Barcelona legend Messi moved to 86 international goals, two more than Hungary legend Ferenc Puskas, now leaving him fourth on the all-time men's list.
Messi's exploits occurred on the same day as tennis legend Rafael Nadal defeated Casper Ruud in straight sets to win a 14th French Open title and extend his record for most men's grand slam titles to 22.
For Argentina boss Scaloni, there are simply no more superlatives left for him to describe either man's seismic impact on the world of sports.
"I don't know what else to say. It's very difficult; it's like Rafa Nadal, what are you going to say?" Scaloni said. "It is preferable that the journalists speak, that they speak. You have no words left to describe it, and above all, for everything it generates."
The result saw Argentina extend itys unbeaten run to 33 matches, fine form just five months out from the start of the World Cup in Qatar.
Messi had completed his hat-trick early in the second half before scoring twice more in the final 20 minutes.
"He is something unique, and it is a pleasure to have him in this group," Scaloni said. "It is a pleasure to train him, the entire coaching staff. And it's a pleasure his behaviour and how he toasts to his shirt.
"We only have words of thanks. I don't think he's just the heritage of Argentina; he's the heritage of the world, of world football. The day he doesn't play anymore, we are going to miss him."