The airlines do not understand the US immigration law well. Print the following out at this link, and show it to the Airlines and ask them to check this link to verify it if you have the same trouble in the future:
About Visas - The Basics (state.gov)
Visa Validity
My old passport has already expired. My visa to travel to the United States is still valid but in my expired passport. Do I need to apply for a new visa with my new passport?
No. If your visa is still valid you can travel to the United States with your two passports, as long as the visa is valid, not damaged, and is the appropriate type of visa required for your principal purpose of travel. (Example: tourist visa, when your principal purpose of travel is tourism). Both passports (the valid and the expired one with the visa) should be from the same country. When you arrive at the U.S. port-of-entry (POE, generally an airport or land border) the Customs and Border Protection Immigration Officer will check your visa in the old passport and if s/he decides to admit you into the United States they will stamp your new passport with an admission stamp along with the annotation "VIOPP" (visa in other passport). Do not try to remove the visa from your old passport and stick it into the new valid passport. If you do so, your visa will no longer be valid.
