Andre Ayew struck to get the African nation back in it in the second half, but John Brooks headed in the winning goal for the Americans.
Speaking before the game, Murphy, who was working as an analyst for the BBC, talked up the calibre of the 31-year-old Dempsey.
Murphy said: "He's a special player because he's not naturally gifted but he looks [as though he is] because of his application.
"He's a Texan boy and he's got the heart to go with it. He's probably the best trainer I ever played with fantastic application."
Dempsey's goal was the fifth-quickest in World Cup history, but the Seattle Sounders man ended up bloodied after being kicked in the nose by John Boye.
Jurgen Klinsmann's side face Portugal on Sunday in their second group game, while Germany meet Ghana.
Earlier in the day, Joachim Low's side recorded an impressive victory against Portugal, as they ran out 4-0 winners, and the Portuguese will head into the game against the USA without Pepe and probably Fabio Coentrao.
The centre-back was sent off against the Germans for a headbutt on Thomas Muller, while left-back Coentrao injured his groin.