There were no goals in regulation time at the Estadio Nacional, though Gonzalo Higuaín spurned a golden opportunity in second-half stoppage time after being played in by Lavezzi - who came on as a substitute when Ángel Di María picked up an injury in the opening period.
Penalties were required after a further goalless 30 minutes of extra time, and Messi found the net with La Albiceleste's first spot-kick.
However, Higuaín missed from 12 yards before Éver Banega saw his effort kept out by Claudio Bravo, leaving Alexis Sánchez to score the winning kick and ensure Argentina's 22-year long trophy drought continues.
And although Lavezzi feels the team missed the chance to go down in the history books, he is sure that La Albiceleste captain will lead the team to glory some day.
"At some point Leo [Messi] is going to win something with Argentina," he told reporters."We thought this was an opportunity to make our mark, but unfortunately we could not. It hurts a lot.
"Once again we are left with a bitter taste. It hurts because we believed that this was the chance to win something.
"I think that we should continue on this path. We are all very good in terms of the group. We are just missing that little step to achieve something."
Javier Mascherano - who has now been a runner-up at three separate Copa América editions - was disconsolate in the face of defeat.
Following their World Cup final defeat in 2014, he described failure at the final hurdle once again as a '"torture"."I cannot find a way to describe it. Hopefully, in the future Argentina can win," Mascherano told reporters.
"I wanted enjoyment with this shirt like anyone, but I have to suffer.
"I spoke with Carlitos [Tevez]. Three finals of the Copa América and I lost all three. I cannot find an explanation.
"I thank the fans for the support, like a year ago at the World Cup. We could not provide the joy that we wanted. It is a torture.
"It's a great sadness, especially for us who were doing very good things."