Short answer (with thanks to commenters):
Yes, you can travel with the existing visa, and no, you do not need to apply for a new visa. However, you need to be sure that the visa in your old passport has not been invalidated (possibly due to a hole through the complete passport), and take your old passport with you. That question is answered here.
More detail for general cases (not USA only):
Most countries have their visas listed digitally. However, your new passport will have a different passport number, and your visa is linked to this number.
I never had to re-apply for a new visa when I updated my passport. When I applied for a new passport, I requested to receive my old passport back as it still had valid visas. Therefore, I traveled with both the new and old passport: the old passport was invalidated, but its visas were not.
If the department that issues your new passport puts a hole into the complete old passport, it also invalidates its visas. Make sure that doesn't happen, and most likely you will not have any issues.
However, it's always good to call the embassy where your visa was issued, and inform them that your passport number has changed.