you must provide an identity document with a proper date of birth to the authorities if you want them to correct the date, better an original document.
The question is: if you are still in asylum procedure with BAMF or with the administrative court or your asylum procedure has been already finished. In the first case you must inform the BAMF or the court; in the second case it is enough to inform the Ausländerbehörde, because the BAMF has nothing to do with your case any more. In both cases you have to give the proof to the Ausländerbehörde in the place you live or lived before. If you want to stay because of the baby without making a new asylum, you must apply for Duldung in your Ausländerbehörde because of the baby. But you must understand that the baby doesn't give you a full guarantee to stay in Germany.
As you must provide the identity documents for the acknowledgement of paternity (Vaterschaftsanerkennung), the Standesamt or Jugendamt can inform the Ausländerbehörde anyway.
It is a good advice from Meike to take a lawyer or at least to consult a lawyer who specialises in Aliens Law. You can also look for a good councelling center for refugees and migrants in your area.