Water damage on 1st page of passport but US visa is intact. Is US Visa still valid?

[Rj Nieto]

The first page of my passport (the one with my name, pic, DOB, etc) suffered relatively heavy water damage. My personal details are still legible, although my picture looks a bit washed out. However, all the other pages, including my US B1/B2 visa, are perfectly intact (zero damage). I already have a new passport to replace this one.

Question: Do I need to re-apply for a new US B1/B2 visa, or can I just bring both the new passport and the old damaged passport with the intact B1/B2 visa when I travel to the US?