Martinez, who came off the bench in extra time, latched onto a precise through ball from Giovani Lo Celso before lifting his effort over advancing Colombia goalkeeper Camilo Vargas in the 112th minute to spark wild celebrations at the Hard Rock Stadium in Florida.
Argentina's latest title comes on the back of their triumphs at the 2022 World Cup and 2021 Copa America, and gives eight-times Ballon d'Or winner Lionel Messi the perfect sendoff in what is likely his last international tournament.
The win, which saw Argentina move ahead of Uruguay after the pair had been tied on 15 Copa titles, marked Angel di Maria's final international match.
Defender Nicolas Otamendi, 36, is also set to end his international career.
"The truth is, this was written, it was this way," 36-year-old Di Maria said. "I dreamed it, I dreamed that I would arrive to the final and win it and retire in this way.
"I have so many beautiful feelings and I am eternally grateful to this generation and today I am leaving in this way with a title."
South American football's governing body CONMEBOL pushed the start time of the match back three times after video on social media showed security scuffling with fans as they ran across the concourses and attempted to breach the gates of the stadium.
A Reuters witness saw several people being ejected from the stadium by security staff at the venue, which will host matches during the 2026 World Cup.
"It was hard because many had their families outside, they didn't know what was going on," Argentina's Cristian Romero told TyC Sports. "It didn't seem like a final, there was a lot of commotion and a lot of concern."
The tournament had already been marred by a number of organisational issues.
Uruguay's players were involved in clashes with Colombian fans earlier this week, while an assistant referee had to be hospitalised after collapsing in hot and humid conditions while officiating a group stage match.
CAGEY AFFAIR
Colombia started the match by pressing high up the pitch and Jhon Cordoba hit the post in the seventh minute before Jefferson Lerma and Richard Rios had long-range efforts saved by Emiliano Martinez.
Messi appeared to hurt his ankle in the 36th minute after a collision with Santiago Arias and while he got back to his feet after receiving treatment he was clearly hampered for the remainder of the half.
Colombia went on the offensive again in the second half with Arias and Davinson Sanchez coming close to scoring.
Argentina soaked up the pressure well but were dealt a blow in the 66th minute when Messi limped off due to the ankle injury.
"Messi had to leave because of that ankle problem but finally we were able to give him some joy," added Di Maria, who jointly lifted the trophy alongside captain Messi and Otamendi at the presentation ceremony.
They continued to grow into the contest and had a goal disallowed for offside but Colombia held on as the match went to extra time.
The defending champions continued to push forward in extra time, testing Vargas with a shot from Nico Gonzalez before Martinez, the tournament's leading scorer, finally found the breakthrough.
'DESERVED MORE'
The defeat ended a Colombian record 28-match unbeaten streak for Nestor Lorenzo's side, whose last loss also came against 'La Albiceleste' in February 2022.
"It hurts after a tournament that the whole team fought for, it hurts when you compete to win and we felt we deserved more," defender Sanchez said after Colombia's hopes of winning a first Copa America title since 2001 were dashed.
"We wanted to give the victory to a country that deserves it, Colombia deserves everything, and we leave empty-handed, but we will go on."
Colombia captain James Rodriguez was named player of the tournament after notching six assists to break Messi's record of five in a single edition of the Copa America.
