A man who allegedly fired a gun in a bar — and then at cops — was fatally shot by police in Queens after he led them on a wild chase Wednesday night, sources said.

The suspect first fired a shot inside the SBS Bar on Hillside Avenue in Jamaica before bolting from the pub at around 9:30 p.m., according to law-enforcement sources.

The shooter then fired at cops at least once as they chased him and was told to drop the weapon before two officers shot him, sources said.

He was struck in the chest and taken to Jamaica Hospital, where he died. What appeared to be a 9mm Glock pistol was recovered at the scene, the sources said.