Arena's team had just two days to pick themselves up from a sapping defeat by Poland in their final group game on Friday.
They did just that to power past neighbours Mexico with goals from Brian McBride and Landon Donovan.
But American coach Arena admitted afterwards that he feared the match would prove a game too far for his tiring squad.
"It was difficult to get our guys up after our last match against Poland on Friday," he said.
There's been a lot of bad blood between the teams over the
"We only had a short period to recover physically."
Arena made four changes to regain some of their earlier sparkle, saying: "We needed some fresh legs in there today."
But he attributed victory to his players' willingness to dig deep into their reserves of stamina and commitment.
"Our guys left everything on the field out there." Arena said.
"They gave a great effort and our veterans did a great job.
"Claudio Reyna played a position he's not used to and we came through big.
"I'm proud of my guys and it is a great day for US soccer."
The game ended on a sour note when Mexican captain Rafael Marquez was red-carded amid a spate of bad tackles by his team.
Arena said that there is rarely any love lost when the neighbours meet, but that his team would not harbour any grudges.
"It's a rivalry," he said." We know each other and there's been a lot of bad blood over the years.
"But when the game's over we're friends again."
The US now have an extra day's rest before facing Germany in Friday's quarter-final.
And Arena believes that will be enough to revive his minnows for yet more heroics.
He said: "We've got a very difficult opponent in Germany but I think when Friday comes round our team will be ready to play."