Jimmy Fallon -- Hospitalized with Hand Injury ... Taping Cancelled (UPDATE)

Jimmy Fallon is laid up in a New York City hospital this morning after having surgery on his hand ... TMZ has learned.

After taping his show on Thursday ... we know Jimmy did dinner at Upper East Side restaurant Scalinatella. He posed for pics with a table full of female fans.

We're told Jimmy left the restaurant around 9ish ... and went home.

His rep says the injury to his left hand happened early this morning in his apartment. He went to the hospital, and a specialist recommended he have surgery immediately.

Friday night's taping of the 'Tonight Show' has been cancelled, but Jimmy's rep says he's expected to make a full recovery.