The Germans dominated the game with Juergen Klinsmann's team, with their backs againt the wall, creating very few chances.
The breakthrough came after 55 minutes when Tim Howard parried out a Per Mertesacker header but Mueller collected the loose ball on the edge of the area and superbly side-footed into the bottom corner.
The game had been threatened by a torrential downpour in the north-eastern city of Recife on Thursday morning but the heavy rain that hampered the preparations of fans and players alike eased to a drizzle as kickoff approached.
Despite the fact that both sides only needed a draw to qualify there was no shortage of attacking intent in the lively opening exchanges, with Mueller going close early on from Lukas Podolski's low cross.
Both sides flew into challenges in the battle for supremacy in midfield, and US coach Juergen Klinsmann and his German counterpart Joachim Loew will be glad the referee didn't wave the yellow card more often as they head for the knockout stages.
Content to concede possession for long periods, the Americans put in a blue-collar defensive performance, breaking forward when the opportunity arose, but with their superior technique and ball movement, the Germans were always a threat.
The Amercans had several chances of their own but lacked the pace and guile necessary to unduly worry German goalkeeper Manuel Neuer.