The United States, coached by former German boss Jurgen Klinsmann, also went through to the last 16 despite the defeat in rain-sodden Recife. Portugal's 2-1 victory over Ghana was not enough to overtake the Americans in second place.
"It's huge getting out of this group," said Klinsmann after the game as he hailed his side's World Cup performance so far.
The lead-up to this match had focused on suspicions that the Germans and Americans would be happy with a draw that would give both sides an easy path through to the next round.
Klinsmann's friendship with his former assistant German coach Joachim Löw had reinforced the conspiracy theories. But Germany dominated the match in the washed-out Arena Pernambuco.
US goalkeeper Tim Howard repeatedly kept his side in the game for 55 minutes until Müller broke the deadlock.
A well-worked corner saw Howard save from Per Mertesacker's header.The rebound went to Müller on the edge of the penalty area who made no mistake with a powerful first-time shot that flew past the flying goalkeeper into his left corner.
Germany could have had several goals in an entertaining first half in which Howard played the key role.
Müller, playing as a lone striker, had several chances to score including a flying bicycle kick from a cross on the right after just two minutes.
Lukas Podolski was given his first start of the tournament and after latching on to Mesut Ozil's ball on eight minutes he whipped in a drive for Müller, who just failed to connect as Howard dived low to collect.
The United States soaked up the early pressure, and on a rare foray, midfield orchestrator Michael Bradley sent Graham Zusi through on the left for the winger to unleash a drive that sailed just over Manuel Neuer's crossbar.
Podolski made way to Miroslav Klose at half-time, giving the veteran German striker a chance to pull ahead of Ronaldo with a new record of 16 World Cup goals.
After only six minutes Klose connected with Kroos's flighted ball in towards the back post but produced only a glancing header that sailed wide of Howard's upright.
Left winger Brad Davis was replaced on the hour by Alejandro Bedoya, who was lucky to escape with a warning after fouling Bastian Schweinsteiger.
Beckerman was then cautioned for a foul on Schweinsteiger, who was later replaced by Mario Götze.
US hopes were finally raised on the stroke of full time but Philipp Lahm produced a superb sliding tackle to block Bedoya after a counter down the right.
From the resulting corner, Bradley whipped in across that saw Neuer tip over from Dempsey's close-range header.