After just 30 seconds at Monumental de Nunez Stadium on Saturday, midfielder Gregore, one of Botafogo's best players, hit the head of Fausto Vera with his foot and was given a straight red card.
"It was so hard to have one less so early in a final," Botafogo defender Alexander Barboza said.
"But Gregore deserves this title as much as we do. We made history and that won't ever be erased."
Winger Luiz Henrique scored first in the 35th minute from close range, and was fouled for the penalty converted by Alex Telles in the 44th.
Eduardo Vargas headed Mineiro's only goal in the 46th while Junior Santos, the top-scorer in the Copa Libertadores with 10 goals, capped Botafogo's historic night with the third in added time.
Mineiro had chances to equalise through Deyverson in the 53rd, Hulk in the 56th, and Vargas in the 86th and 88th, all from close range.
Brazil was assured of a sixth consecutive Copa Libertadores title and 24th in history, just one less than Argentina.
It was also the third straight title for Rio de Janeiro clubs, after Fluminense in 2023 and Flamengo in 2022.
Botafogo, the last of the four big Rio clubs to win the Copa Libertadores, earned prize money of $US23 million ($A35 million) and a spot in the Club World Cup in the United States next year.
"We just failed at getting some advantage at having one more player than them," tearful Mineiro defender Rodrigo Battaglia said.
"It was a very hard day for all of us."
Botafogo coach Arthur Jorge joined Portuguese compatriots Jorge Jesus (Flamengo, 2019) and Abel Ferreira (Palmeiras, 2020 and 2021) as European coaches with a Copa Libertadores title.
Botafogo also gave American owner John Textor his biggest trophy yet. Textor has been subject to criticism after unproven allegations about match-fixing when Botafogo squandered a 13-point lead last year and missed out on the league title.
Next week, his team could win their first Brazilian title since 1995.
Botafogo were relegated from the league in 2020 but have risen to prominence again with Textor's investment. He was part of a wave of foreign owners who came into Brazilian soccer after a 2021 law change paved the way for private investors.
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