The Magpies have released a statement in which they said they were "embarrassed and appalled" by the behaviour of those involved in the disturbances after the 3-0 Barclays Premier League defeat at St James' Park.
It said: "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."
The statement continued: "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."
A total of 29 arrests were made after pockets of violence broke out in the city centre after the game. Video footage showed some supporters throwing bottles at police, while one man was photographed apparently aiming a blow at a police horse.
Bud, of West Yorkshire Police's mounted section, was back home in his paddock on Monday safe and well after the suspected attacker was wrestled to the ground by officers, and Northumbria Police say he has been bailed pending further inquiries.
The West Yorkshire Police Horses Twitter account have tweeted a picture of Bud, saying he was not injured during the scuffle.
It said: "Bud is safe in his stable and appears fine. All our horses are fit and well, resting and relaxed."
Three police officers were injured during the clashes as sirens rang out across the city centre in the immediate aftermath of the game.
Chief Superintendent Steve Neill, of Northumbria Police, said on Sunday: "The vast majority of fans were well behaved today and enjoyed the Tyne-Wear derby for the great occasion that it is.
"Disappointingly, there was some disorder involving what appear to be Newcastle United supporters following the match in the city centre, including an incident at Central Station.
"Disruptive behaviour of any kind is not tolerated. Inquiries will take place into these incidents and action will be taken against all those involved."
A crowd of 52,355 packed into St James' to see Paolo Di Canio's Sunderland upset the odds and record their first derby win on Tyneside for 13 years to boost their chances of avoiding relegation. Defeat left Newcastle, who bowed out of the Europa League on Thursday evening, only two points better off on 36, five clear of the drop zone with just five games to play.
British Transport Police (BTP) announced they made 36 football-related arrests including for violence and racist abuse during a number of outbreaks of football disorder over the weekend, including at Newcastle. Of the four injured officers, three were from the force including a female officer who was taken to hospital after a brick was thrown at her shoulder.
Following the second FA Cup semi-final, four Manchester City supporters were arrested at rail stations for offences including racially-aggravated public order, assaulting a train conductor and possession of drugs. Four Chelsea supporters were arrested at Wembley Park for offences including common assault and racially-aggravated offences.
On Saturday, BTP officers arrested six Millwall supporters for offences including affray, theft, racially-aggravated and public order offences and being drunk and disorderly.
In other incidents, 11 Watford fans were detained at Kings Cross.
BTP Chief Constable Andrew Trotter said: "This was a very busy and, at times, very testing weekend of football policing.
"I'm extremely proud of my officers, who, alongside colleagues from local police forces, stood firm in the face of some truly difficult situations. This was most apparent in Newcastle on Sunday, where three BTP officers were injured in the line of duty. None, I'm glad to say, were seriously injured.
"This weekend saw at least 36 football-related arrests on the rail network, and the troublemaking minority were dealt with swiftly and robustly. However, the scenes at Wembley Stadium on Saturday and in Newcastle on Sunday serve as a firm and sobering reminder that there is still much more to be done in tackling football-related disorder."