Procedure for changing the date of birth in the passport urgently

To change the DOB in the passport you must apply for 'reissue of passport' which I believe you must have done as they have given you a date. You might had submitted the copies of the desired documents also with that. Now the important thing is that they will ask you some original documents to prove the point and will verify it. They may even ask some additional document also. The reason for that is that whenever there is a change of DOB is involved then a doubt is created in the mind of the verifying authority about it and they want to be doubly sure before approving that.
Anyway what I suggest is to keep with you if possible some other supplementary documents which has the correct DOB. For example some school leaving certificate or municipal birth certificate. That would help you immensely.
There is a 'document advisor' link in the homepage of the passport site (https://www.passportindia.gov.in). If you have not gone through that then just visit and see the details there.

The change of date of birth on an Indian passport can be done only if
1. If you request for a change of date of birth within 5 years after the issue of the passport.
2. If the passport holder is a minor and the wrong date of birth is mentioned in the passport, he/she can apply for correction after attaining 18 years. In this case, 5 years clause is not applicable,
You have to apply online for a reissue of your passport and when the appointment is given you have to attend the Regional Passport Office. When you are filling out the application you have to write your correct date of birth and proof of correct date of birth is to be produced.
The certificate of birth issued by the municipal authority or the office of the Registrar of Births and Deaths or other designated authority is a valid proof of date of birth. In the case of applicants born on or after 26th January 1989, it is mandatory to furnish this proof.
As a matter of fact, we should check the date of birth as soon as we receive a certificate and if there is any difference we should apply immediately. As per rules the date of birth of a minor can't be changed if she has not attained the age of18 years. I don't how the passport authorities can help you.

There is an error in your question. At one place, you have mentioned the correct DOB as 22 March 2006, but at the same time, you have said that - ' I have other documents like her Aadhar card, birth certificate, and 10th certificate which has her correct DOB (20 March 2022).'

However, I am answering this question taking the correct DOB as 22 March 2006. That means your daughter is still a minor and has not yet completed 18 years.

The Date of issue of the passport is also relevant in answering this question which is not mentioned. Have you gone for a reissue of the passport after five years of its issue?

The Date of birth in the passport of a minor can be corrected anytime when the passport holder is a minor or when they apply for a reissue of the passport after becoming a major.

Kailash Kumar, the date stands corrected now.