The Briton, whose championship-leading team mate Nico Rosberg will be on pole position for his home race, qualified 16th on Saturday despite spinning into the barriers early on.
However, he gained a position because Sauber's Mexican Esteban Gutierrez carried over a three-place demotion from the previous race.
But Mercedes said Hamilton would also have to take a mandatory five-place penalty for the unscheduled gearbox change overnight.
"We can confirm that @LewisHamilton's gearbox will be changed for the race owing to accident damage incurred yesterday. 5 place penalty," the team said on their Twitter feed.
Hamilton's crash, in the first phase of qualifying, was caused by a front brake disc failure and Mercedes indicated on Saturday he would change to a different brand as a safety precaution.
The governing FIA said that would not carry a penalty.
"We can now confirm @LewisHamilton will be starting from 20th on the grid this afternoon. Buckle up!," the team said in a later Twitter post.
Hamilton is four points behind Rosberg at the halfway point in the season with nine races remaining after Hockenheim.
(Reporting by Alan Baldwin, editing by Patrick Johnston)
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