Defender Jonathan Bornstein headed in the equaliser in stoppage time to shock the Costa Ricans, who had been on the verge of advancing to the World Cup finals.
"We always talk about when you step on the field you want to show people what you're all about," said Bradley.
"Tonight was not the way you draw it up on the board before the game. But we showed our mentality, our spirit, and we're very proud of that... It was a very determined, strong effort."
When Bornstein's goal slammed into the back of the net off a corner kick in the 95th minute, the crowd of 26,243 on a damp and chilly night in Washington roared its approval.
"We were pushing extremely hard that whole second half," said Bornstein. "We threw a lot of numbers in the box. For corner kicks, I usually stay in the back.
"But for that one, I said 'I'm getting in there.' It was a last-shot effort. I just snuck in there. I found myself wide open."
The Costa Ricans stood in stunned silence, many of them holding their head in their hands. Others pulled their shirt over their head.
By virtue of the draw, Honduras, who won 1-0 in El Salvador, took the third automatic qualifying spot in CONCACAF leaving Costa Rica to face Uruguay over two legs.
The United States, which had already qualified for the World Cup, finished first in the group.
Costa Rica went 2-0 up in the first half through goals from Bryan Ruiz and had looked certain to secure their spot in South Africa along with the United States and Mexico.
Michael Bradley scored in the 72nd minute to the trim the U.S. deficit to 2-1, setting the stage for Bornstein's heroics.
"It shows a lot about our team, what we're all about," Bornstein said of the second-half rally.
