Messi captained his national side and was proud of their display in the tournament, which culminated in victory over Chile in the third-place play-off.
La Albiceleste fell short in a major competition yet again, but Di Maria was inspired by what Messi had to say.
"He said some beautiful words when we were eliminated by Brazil," Di Maria said. "Messi told us he was proud of the team we had built. That we had not been together much as some had not been called up [to the senior team] before. Yet, it felt as though we had all been together for many years.
"He said that we had all rowed in the same direction from the first day. That he was very proud of the youngsters for their commitment to this jersey and if they were there and gave what they gave at the Copa America, it was because they more than deserved to be there.
"Once he finished, everyone was in tears because it touched everyone's hearts, especially the youngsters."
Messi has been unable to replicate his Barcelona form on the international stage with Argentina, and critics have accused him of lacking the passion required to skipper the side.
But Paris Saint-Germain attacker Di Maria claimed he has seen a different side to Messi this year.
"There was a lot of criticism that he didn't sing [the national anthem], that he didn't talk," Di Maria said.
"This Copa America was different. He proved it. What made me happier was to see how he spoke in front of the group, how he spoke to reporters, what he said. The important thing is that Leo is like this. I like this Messi."