The 36-year-old  was travelling on the Metrolink service at about 06:50 BST on Saturday near Beswick in Manchester when the argument broke out.
The offender "sliced the victim's neck with a knife" before running off at the Velopark stop, officers added.
The victim was taken to hospital where his condition is stable. Police are appealing for details from witnesses.
Insp Steve Griffin, from Greater Manchester Police, called it a "vicious" attack, adding: "It is only good fortune that prevented this man from suffering potentially life-threatening injuries, because if this knife had cut an artery I would dread to think what the result could have been.
"We know there were a number of people on the tram at the time, and we need them to come forward and tell us what they saw."
The offender is described as black, aged between 35 and 45, about 5ft 4in tall with a shaven head.
He was with a woman, who was white, wearing dark hair in a ponytail and black clothes.
A man was stabbed in the neck on a tram after a row with a passenger, police said.