Julian Alvarez opened the scoring in the 13th minute, thanks to some terrific work from Lautaro Martinez, who was able to snake his way into the box.
Martinez beat multiple defenders to get to the touchline and drag one back along the edge of the six-yard box for a simple tap-in.
Argentina continued to create half-chances throughout the remainder of the first half, and it was more of the same to begin the second 45 minutes.
Martinez went close after a slick one-touch pass from Alexis Mac Allister gave him some space inside the box, but he pushed his shot wide of the far-post.
Lionel Messi was subbed on in the 56th minute, and just 10 minutes later he had his first chance.
This was after a one-two on the edge of the area allowed him to break into the left-hand side of the box.
But his shot from a sharp angle was easily thwarted at the near post.
However, fans in attendance would get what they came for in the 87th minute, when Messi picked up a deflected pass just outside the penalty area.
He fired his left-foot shot hard and low across his body into the bottom-right corner.
Just two minutes later Argentina were awarded a free kick in the same spot, and Messi again converted, this time around the wall into the bottom-left corner.
With this result, Argentina are now unbeaten in their past 35 games dating back to their loss in the 2019 Copa America semi-final against Brazil.
Italy hold the record at 37 games without a loss —- which they set just last year.