"They've shown that in the group games, and I thought the way they played was fantastic. I'm so proud of what we've achieved, individually and collectively, and how we've represented the country."
Colombia took an early lead against the run of play when Andres Reyes glanced a 10th-minute header inside the far post from a free kick.
New Zealand equalised in the 35th minute, Liberato Cacace powered a low ball into the box for Elijah Just to expertly turn home.
New Zealand's best chance of taking the lead arrived just past the hour when Just got on the end of a lofted pass from Singh and lifted the ball over goalkeeper Kevin Mier but a covering defender cleared the shot.
Extra-time brought more drama as both teams had chances to win and New Zealand were indebted to Woud, who pulled off an exceptional save from one of the last kicks of the game to keep the scores level.
That meant the tie would be decided from the penalty spot and substitute Matt Conroy was the unlucky one, seeing his decisive effort saved for Colombia to finally triumph after Woud's double-save heroics.
Buckingham had words of comfort for Conroy and the rest of his crushed squad as they look to bounce back from the nature of their World Cup elimination.
"There's going to be some devastated players here but they will reflect upon the last four or five weeks and realise the impact this has hopefully had on football in New Zealand," he said. "We've come here and outplayed teams. We have good players and you've seen that again [against Colombia]. It's a horrible way to exit the tournament, especially when you're that close to creating history, but I couldn't be prouder of every single one of them."