The former Daily Mirror editor wrote that the incident was especially uncomfortable as he had made his feelings about his "unfortunate personality transplant" known on Twitter.
"You can imagine how I felt when I sat down in my favourite New York restaurant, Ralph Lauren's Polo Bar, ordered a fine bottle of vintage claret, sighed with almost indecent pleasure at the gastronomic delights heading my way... then turned to my left and spied Cleese at the very next table. Literally two feet away.
"For one tiny nanosecond, our eyes locked in mutual shock, then equally mutual withering contempt. For the next two hours, we avoided all form of contact – physical, verbal, visual. It was magnificently British."
Cleese wrote in response: "Piers Morgan writes that I didn't recognise him in a restaurant in New York. I did. I just didn't want to speak to someone I truly detest."
This sparked a flurry of exchanges between the two:
This isn't the first time the divisive columnist has had a high-profile argument.
Morgan was also involved in long-lasting feuds with TV presenter Jeremy Clarkson and TV personality and Private Eye editor Ian Hislop.