I had posted this on the IMDB forum, and so far have gotten zip for an answer, which bothers me because those cats are really sharp, and it’s leading me to believe I’ve imagined the whole thing. Anyway, here’s what I remember:
I thought this was “Amazing Stories,” but maybe not. Anyway, it was somewhat light-hearted, and I’m thinking the original invasion was broadcast by radio (like "War of the Worlds), and at the end, we find out there there is indeed no Martian invasion when one of the characters makes the comment “Now for the invasion from Jupiter! (or Saturn).” Could have sworn it was Harvey Korman, but I didn’t see him in any anthology in that time frame.
I believe this is “Panic”, from Perversions of Science (the follow-up to HBO’s Tales from the Crypt, inspired by EC’s science fiction titles). You can watch the whole episode here.
There are some of these elements in the TZ episode, “Will the Real Martian Please Stand Up?” It takes place in a diner, where a busload of people have been stranded.