"We had something like 27 chances in the game and we deserved our win. We created a chance after just two minutes and it was fantastic goalkeeping from Tim Howard that kept us out," Wilmots said.
"We saw a lot of football from Belgium tonight. I'd say we played a good game, we were brave and we are into the quarter-finals."
Belgium had been taken to task for failing to live up to expectations at the tournament in Brazil even though they won all of their opening round matches.
One of those heavily criticised was striker Romelu Lukaku, dropped for the last-16 match against the Americans at the Fonte Nova arena in Salvador, but whose entry at the start of extra time finally turned the tide Belgium's way.
"It was right to criticise him but I told him that the knockout phase would belong to him, he would have his day," Wilmots said.
But the heroics of Howard and the plucky fight of the Americans meant the contest was not settled until after two hours of furious football.
The Belgium coach said he suffered near heart failure as his side failed to take any of numerous chances in the 90 minutes, then scoring twice to seemingly settle the game before allowing the Americans back into the contest in a edge-of-the-seat finish.
"Usually at this level of football when you miss that many chances you don't advance, even if you play fantastically. But we were able to show mental strength and a physical level to see us through," he said.
"I can believe it was a brilliant game for the spectators to watch but it was no good for my heart. I said to the players after, 'don't do that to me again'."
Belgium will face Argentina on Saturday in Brasilia in the quarter-finals of the tournament.
- Sports & Recreation
- Tim Howard
- Romelu Lukaku