Kevin De Bruyne and substitute Romelu Lukaku scored early in extra-time as Belgium scraped home to win 2-1 against the USA in Salvador and set up a quarter-final showdown with Argentina.
Substitute and 19-year-old winger Julian Green - the USA's youngest ever World Cup player - pulled a goal back for Jurgen Klinsmann's men with 13 minutes remaining in the second additional half of their last 16 clash and it spurred the team into action at the Itaipava Arena Fonte.
But despite the USA's best efforts, they could not find that elusive second goal to send the match to penalties, much to the disappointment of Howard.
"It's heartbreaking," Howard said post-game. "We gave everything, I don't think we could have given anymore. What a great game of football.
"We left it all out there. We got beaten by a really good team. They took their chances well.
"It's heartache. It hurts."
Howard added: "The dream falls short but, like I said, this is an incredible group and we'll never forget this night."
"It's my job, that's what I sign up to do," Howard said. "It's part of it in these big games against top quality competition.
"The levee's going to break at some point if we continue that.
"Hats off to Belgium, they were fantastic but we gave a valiant effort."
"They brought big Rom on," said the Everton shot-stopper. "He was a handful.
"He ran at us, scored a goal, created chances for other guys.
"That's what you want from your substitutes. Julian came on and did the same thing for us.
"Big Rom changed the game. That's what he does."
"You get to this point and these games are always about a play here, a play there," said Bradley.
"Changes were going both ways. Right at the beginning of overtime we gave up a goal to go down 1-0 and probably then got a little sloppy to give up another.
"Having said that, we kept brave, we kept fighting to get a goal.
"Then after getting back to 2-1 we're probably unlucky not to get the equaliser at the end."
