Diego Forlan had given the Spanish side the lead before Simon Davies equalised soon after and the game appeared to be heading for penalties before the former Manchester United striker pounced again in extra-time.
Hodgson told Five: "We have to give them real credit but I thought we played well. It looked for all the world to be heading for penalties which is a bit of a lottery but we would have been happy to take that lottery.
"I thought we looked comfortable for long periods of the second half and extra time.
"All through the season they have performed against the odds and pulled out performances.
"You can't be happy after any sort of defeat, even after extra time, but I don't have a word of criticism for the team."
Hodgson was loath to describe the Cottagers as the best squad he has ever coached, given his long and illustrious career, but he did pay tribute to their spirit.
"They are certainly best in terms of attitude, desire, organisation and commitment," he said.
"I wouldn't like to say the best because that is negative to other squads I have had.
"They have every reason to be very proud of the season and tonight even if they are not lifting the trophy.
"We wanted to win and that shone through in everything we did out there.
"Everyone realised we gave our best.
"I thought we kept Forlan and (Sergio) Aguero pretty quiet considering how good these two players are, but we didn't keep them quiet quite long enough."
Goalscorer Davies admitted he might appreciate the enormity of the team's achievements once defeat has sunk in but it was still too raw immediately after the game.
He told Five: "Getting so close to penalties we were hanging on a little bit. We were hanging on and looking to try our luck there. It's a shame to get so close but once we get home after a few days we'll feel like we achieved something but now it is very painful.
"Scoring in a big final I'll be very proud in years to come but now it is really hard.
"We were so close but we must congratulate Madrid."