The 21-year-old Borussia Dortmund player was taken off in the 66th minute of their 1-0 victory over England on Wednesday after a knock on his thigh.
"He left the team camp this Friday morning," the German football association said on Friday, with the team set to leave for Baku later in the day.
Weigl's departure comes a day after striker Timo Werner was ruled out with a thigh muscle injury.
Germany will have several players back, though, including striker Mario Gomez and Mesut Ozil, who did not take part in the final training session on German soil.
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The world champions are top of their qualifying Group C with four wins out of four matches and a five-point lead.
(Reporting by Karolos Grohmann; Editing by Gareth Jones)
