With 600 soldiers due to take part in the evacuation, Germany hopes to be able to get more people out in the coming hours.
Flights resumed early on Tuesday after evacuations were suspended the day before as people crowded the runway trying to flee after Taliban insurgents seized power.
Germany is waiting for US permission to fly a second aircraft in from Tashkent, which it is using as a hub.
Kramp-Karrenbauer hoped the airport could be kept open for several days so as many people as possible could be evacuated.
"Those are not only German citizens, they are also local staff, at-risk people. We will also support other countries," she added," she said.