The passport which has my u.s visa expired recently (I got a new passport). However, the old passport bio page is now partially damaged (bottom part with the signature and Expiry & issuse date tore out). The visa is intact with absolutely no damage. Do I need to apply for a new visa in my new passport???
You do not need to apply for a new visa. Make sure to take the old passport (with the visa) with you when you come to the US.
Source: US State Department, which says on this page:
Unless canceled or revoked, a visa is valid until its expiration date. Therefore, a valid U.S. visa in an expired passport is still valid. If you have a valid visa in your expired passport, do not remove it from your expired passport. You may use your valid visa in your expired passport along with a new valid passport for travel and admission to the United States.
The Answers to this Travel.SE question add that the two passports must have been issued by the same country, and must be of the same type.