Officers said they were treating the attack, at 04:00 on Saturday, as attempted murder.
It followed an argument between two men which broke out on North Bridge, near the junction with High Street.
The 36-year-old was stabbed in the torso and taken to the Royal Infirmary of Edinburgh. The attacker ran off in the direction of Princes Street.
The suspect is described as black, in his 40s or 50s, about 6ft 5ins tall with a large build. He had short, greying black hair and spoke with an African accent.
He is believed to have been wearing a shirt and jeans.
Police have appealed to anyone who can help identify the man to come forward.
Det Insp Grant Johnston, of Gayfield CID, said the victim managed to make it as far as the Balmoral Hotel after the attack, before the alarm was raised.
"We're treating this attack as an attempted murder and are pursuing every line of inquiry," he said.
"Today, I urge anyone who may have witnessed this incident or who recognises the description of the male suspect, to please contact police immediately.
A 36-year-old man is in a serious condition in hospital after being stabbed in Edinburgh city centre.