USA were up after just 32 seconds as Clint Dempsey stormed into the box, cut inside and finished brilliantly past Adam Kwarasey in the Ghana goal.
After sustained pressure, Andre Ayew finally found the equaliser for the Black Stars on 81 minutes, yet their joy was short lived as American substitute John Brooks headed home with just four minutes left on the clock.
Appiah acknowledged that his opponents were a tough nut to crack, yet felt his side gave it their all throughout the 90 minutes.
“What I can say is that it was a very tough game," the Ghana manager said at the post-match press-conference.
"Playing at this high level, any little mistake can cost you dearly.
"We didn't deserve the first goal against us and, as for the game itself, we did possess and create a lot of chances.
Appiah was left bitterly disappointed at his side’s failure to convert just one of their 21 attempts at goal.
"Unfortunately we could not take our chances and the US took their chances,” he said.
"Any loss of concentration can cost the team big time. I believe the first goal unsettled us a little bit but I never expected it to end this way."
Ghana next face Germany, who were comfortable 4-0 winners over Portugal, in their next group encounter on Saturday, 21 June.