Hummels, returning to the side after illness, nodded home a Toni Kroos free-kick in the 12th minute at the Maracana on Friday and that ultimately proved enough to book a semi-final date against Brazil or Colombia in Belo Horizonte next Tuesday.
Didier Deschamps made two changes to the France side that began the 2-0 win over Nigeria. Mamadou Sakho replaced Laurent Koscielny at the back, while Antoine Griezmann was preferred to Olivier Giroud on the left side of the attack.
Germany needed extra-time to get past Algeria 2-1 on Monday, and coach Joachim Low’s starting line-up showed three changes from that game. Centre-back Hummels was preferred to Per Mertesacker, while veteran striker Miroslav Klose replaced Mario Gotze in the forward line.
After an early half-chance had come and gone, Karim Benzema was looking to get onto a pass from Griezmann only for Hummels to come in and intercept.
And when Paul Pogba fouled Kroos on the left touchline soon afterwards, the same Germany player sent in a free-kick which Hummels met with a header in front of Raphael Varane, directing the ball past Hugo Lloris for the opening goal.
When Klose went down in the box in the 24th minute under a challenge from Mathieu Debuchy, it looked like Germany might be given a penalty – but Argentinian referee Nestor Pitana turned down their appeals.
On 34 minutes, Benzema’s hooked effort was diverted over for a corner, and the France striker had penalty appeals of his own just before the break when he claimed Hummels had committed a handball appeal, but again Pitana was unmoved.
Khedira was the first player booked when he brought down Griezmann on the halfway line eight minutes after the restart.
After Andre Schurrle replaced Klose, Deschamps threw on Koscielny and Loic Remy in place of Sakho and Yohan Cabaye respectively, but still France could not find an equaliser.
After Gotze had come on for Ozil, Deschamps made his final throw of the dice as Giroud was brought on in place of Valbuena.
In the 87th minute, Germany was given a great chance on the break but Schurrle wasted the opportunity after being found by Muller.
In the final minute of stoppage time, Benzema’s shot from the left side of the area was punched away one handed from under the crossbar by Manuel Neuer as Germany prevailed.