Extra-time goals from Kevin De Bruyne and Romelu Lukaku helped Belgium to a 2-1 win, with Julian Green making for a nervous finish thanks to his goal in Salvador.
USA goalkeeper Tim Howard starred with multiple saves in the round of 16 clash in Salvador.
Wilmots believes his team deserve their quarter-final place and praised his players for their fighting qualities.
"We had at least 15 scoring opportunities, so we deserved to go through, even though it was scary to the very end," he told reporters.
"The players fought hard for this qualification and dug deep. All of Belgium can celebrate this quarter-final."
Lukaku had a major impact after coming on at the end of regulation time, setting up De Bruyne before scoring himself.
It was Lukaku's first international goal since a June 1 friendly win over Sweden, and Wilmots said the 21-year-old was well-prepared for the USA clash.
"Romelu was going through a difficult time and had received a lot of criticism after recent games, but I had prepared him well for this match," Wilmots said.
"This is the life of a striker, I’m afraid, as you are judged on goals."
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