Cristiano Ronaldo's goal in extra-time proved decisive for Jose Mourinho's men, and the Spaniard was thrilled to capture his first piece of silverware since joining the club from Liverpool in 2009.
"No doubt there were many players eager to get their first trophy of the season," Alonso told reporters after the game.
"We wanted to win a trophy and I'm very happy for the club."
However, the 29-year-old did not get entirely swept away after the game as he acknowledged the gallant effort put up by their Catalan counterparts.
"They [Barcelona] played very well but this is a time for us to enjoy the moment," he said.
Alonso has already made plans for where the trophy will be taken to celebrate with the club's supporters.
He said: "For the first time it [his move to Real Madrid] feels worth it and we should celebrate this at [Plaza de] Cibeles."
Real Madrid will face their arch-rivals again in the Champions League semi-finals, with the first leg at the Bernabeu on April 27 and the return fixture at Camp Nou on May 3.