This topics about changing the name on the Indian Passport have been discussed in depth in several articles here at Happy Schools. Start from the following article and follow other links.
This is a common problem in India that there is no importance given for separate first name and Surname. Even my son had a similar problem and we realize this only at the time for applying his visa & no time was left to get passport corrected.
It’s always better if one gets his passport corrected before applying for US Visa. Thats much easier.
But in case if someone cannot get his passport corrected before getting US Visa, there is a procedure with CGI (Counselor General of India) in USA. One can go through their website and follow the procedure for change of name (Splitting).
Its not very complicated, one can download all the forms to be submitted (from the CGI website), fill the form, get them attested by some attorney in US, get a draft for requisite fee from the bank and mail all the documents, along with all original passports (old & new) to CGI.
See that you comply with the checklist provided at the CGI website. My son, who is studying in Los Angeles, did the same and got a new correct Passport from CGI, San Francisco through mail within less than a fortnight.
Later on, one can continue on his old valid visa in the previous passport, provided both the passports are stapled together and get the new visa on new passport whenever he comes to India.
Please note that one can get a new visa from the same embassy which issued first visa. This change in the name must be done at the earliest possible so that getting SSN, DL etc are done with proper name only.