Violence broke out between Newcastle fans and police after the final whistle had gone, with 29 arrests being made, with three police officers suffering injuries.
Sunderland's victory was their first at St. James' Park for 13 years and their biggest derby win since 1979.
The Newcastle statement read: "We were embarrassed and appalled by the behaviour of a minority of so-called fans who last night were involved in disturbances and disorder in Newcastle city centre after the match.
"Newcastle United have been working with Northumbria Police, and will continue to do so, in order to identify all of the individuals concerned.
"The club will take the strongest possible action against those involved in the disturbances and will impose immediate lifetime bans on all those found guilty.
"These deplorable individuals have no place at Newcastle United and bring shame on the club and the vast majority of its proper, law-abiding, fans."