Enter USA with valid H1B visa on expired passport & renewed passport

[h1b_candidate]

Hello, I have a H1B visa on my passport which is valid for next two years. However, since the passport is set to expire in coming months, I have gotten my passport renewed now.

My plan is to travel to USA with the new passport and valid H1B visa printed on old/expired passport as well as copy of approved I797 petition.

I want to know if this is acceptable from immigration check point of view. Or do I need to take a new H1B visa on my new passport?

[newacct]

That is perfectly fine, as long as the new passport is issued by the same country and of the same type (e.g. ordinary passport, official passport, etc.) as the old passport, and you didn't change your name or anything.