Carlos Protesoni and Kike Garcia scored two goals in two minutes in the final stages but Real Madrid held on.
Colombia, despite having Daniel Munoz sent off just before the interval and facing plenty of Uruguay pressure, hung on for a famous victory to the delight of their massive following of fans.
Kylian Mbappe joins Real Madrid after the club won the La Liga and the UEFA Champions League last season.
Eder Militao and Douglas Luiz missed penalties for Brazil, after the game finished 0-0 in 90 minutes.
USA were knocked out after a 1-0 loss to Uruguay, as Panama qualified ahead of them after a win over Bolivia.
A moment of madness from Timothy Weah left the USA's Copa America campaign hanging by a thread on Thursday as the hosts crashed to a shock 2-1 defeat to Panama, while Uruguay hammered Bolivia 5-0.
Both teams drew 3-3 in the first leg at the Spanish capital last week following Federico Valverde's late-equaliser for Real Madrid.
Real Madrid have a difficult balancing act to pull off ahead of the Champions League quarterfinal second leg at the Etihad next Wednesday, after a thrilling 3-3 draw earlier this week.
Alex Meret's unfortunate own goal handed Real Madrid a 3-2 win at Napoli on Wednesday which moved the Spanish giants top in Champions League Group C.
Real Madrid came from behind to beat Real Sociedad 2-1 and climb back to the top of La Liga on Sunday, maintaining their perfect start to the season.