I think I may have found it on the wikipedia page for nanotechnology in fiction:

> The 1984 novel Peace on Earth by Stanislaw Lem tells about small bacteria-sized nanorobots looking as normal dust (developed by artificial intelligence placed by humans on the Moon in the era of cold warfare) that has later came to Earth and are replicating, destroying all weapons, modern technology and software, leaving living organisms (as there were no living organisms on the Moon) intact.

It's not a perfect match for your description, nor is it from the 60s, but it was the closest match I could find. Is that the book you were thinking of?