The contest between the two European powerhouses lacked spark and Hummels's 13th minute strike spelt doom for France at the iconic Maracana stadium.
Germany keeper Manuel Neuer was also outstanding under the bar as he made two spectacular saves in each half to deny France the equaliser.
Neuer also made a last-second save when he pushed Karim Benzema's strong left-foot strike from the left to send his side into the World Cup semi-finals for fourth successive time.
The Germans were clearly the more dominating side and they were rewarded for their effort as Mats Hummels headed home a superb free-kick from the left taken by Toni Kroos.
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Hummels not only jumped well but also out-powered French defender Raphael Varane, nodding home the lead in the 13th minute.
The Germans continued to enjoy great possession thanks to to some superb work by their midfield trio of Toni Kroos, Sami Khedira and Bastian Schweinsteiger.
Germany also survived the resultant corner as Mahadou Sakha headed over the cross bar.