Dempsey fired the Americans ahead inside the opening minute on Monday before Andre Ayew looked to have secured a draw for Ghana in the 82nd minute.
"The result shows our character," Dempsey told FIFA.com. "We got the goal the goal early in the game and that caused us to be a little bit more defensive.
"I thought Ghana played really well, but I thought that we didn't help ourselves at times in terms of keeping possession.
"When they got the equaliser, I thought that we might get a point out of the game, but credit to our team, we kept fighting and we got the goal. We didn't play our best, but we ground out the result."
Dempsey and his teammates return to action on Sunday when they face Portugal.