Chile met Argentina in the final at the Estadio Nacional stadium on Saturday, but 120 minutes of football failed to produce a goal so the game went to penalties.
Jorge Sampaoli’s men emerged as 4-1 victors from the shoot-out, with Bravo denying Ever Banega from the spot after Gonzalo Higuain had blazed over before Alexis Sanchez converted the winning kick.
The triumph meant Chile got their hands on the trophy for the first time in their history, and the goalkeeper was delighted to have reached one of his career goals.
"I always dreamed of lifting the Copa one day," Bravo told reporters after the match.
"It was the moment. We had to leave the negative history behind. This is dedicated to all Chileans."
Chile were eliminated from the World Cup in Brazil last year in a penalty shoot-out with the hosts, and Bravo believes that held them in good stead this time around.
"We had the experience from the World Cup. I told my team-mates that they would miss two penalties and we were going to win," he added.
Teammate Arturo Vidal – who was awarded the Man of the Match award for his role in the victory – also dedicated the title to the nation’s public and stated that Chile were deserved winners.
"The Chilean people needed a win, something as beautiful as being champions of the Copa America is what I think this country needed," Vidal said after the game.
"Now we will fight to the maximum for the next World Cup, but today we took an amazing step and this generation deserved it."