The scheduled one-off gig was scrapped last week after Morrissey collapsed following a show in Boston which he blamed on catching "a horrendous cold" from his support act Kristeen Young.
Sir Cliff, who has had a handful of hits in the United States, said the June 21 show was to thank his "loyal fans" and would include a question and answer session as well as performances of hits including Devil Woman and We Don't Talk Anymore.
He said: "I know that many fans were disappointed about the Morrissey concert being cancelled and I was disappointed as well.
"I've decided I'm going to go anyway to New York and I'm going to put on an evening. While it won't be the show I had planned to do with Morrissey, it will be a really nice, intimate evening with my fans and I'm looking forward to seeing everyone there."