West Ham will issue life bans to any fans found responsible for attacking the Manchester United team coach before Tuesday's game at Upton Park.
A window on the bus was smashed and the kick-off delayed by 45 minutes as a result of the trouble.
"We are aware that there were some supporters outside the Boleyn Ground who didn't act in an appropriate way," a club statement read.
The Football Association also condemned the "unsavoury incidents".
It added: "We will work closely with both clubs and the Metropolitan Police to fully investigate these matters."
Bottles were also thrown at Manchester United goalkeeper David de Gea during the match, while another fan invaded the pitch to confront the Spaniard.
West Ham eventually won 3-2 in what was their final game at the ground before moving to the Olympic Stadium.
The club statement continued: "It was an extraordinary night full of extraordinary moments in front of extraordinary fans, 99% of whom behaved impeccably and were a credit to the club.
"We have already had thousands of tweets and emails from fans to say how proud they were to be a part of such a special evening in West Ham's history."
However, the club said the behaviour of some fans was "not acceptable" and insisted they "will work with the police to identify those responsible and ban them for life".
West Ham vice-chairman Karren Brady used Twitter to condemn the attack on Manchester United's bus.
She added: "The events in the stadium last night were incredible and it would be a shame if the actions of a mindless few outside overshadowed them.
"Their behaviour was unacceptable and won't be stood for. But my overriding memory will not be of them but of a fitting farewell to the Boleyn."
Footage filmed on the coach by Manchester United winger Jesse Lingard showed the players shouting, screaming, laughing and hiding on the floor as the coach was hit by objects thrown outside the ground.
"It wasn't nice, the coach got smashed up," said Manchester United captain Wayne Rooney told Sky Sports. "I'm sure West Ham as a club will be disappointed with what the fans have done."
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Manchester United manager Louis van Gaal, whose side missed a chance to move up into fourth in the table, added: "I think the images shall tell everything."
Speaking immediately in the wake of the incident, West Ham co-chairman David Sullivan denied the coach had been attacked.
"There were people around it and people were being pushed in all directions but there was no attack on the coach," he told BBC Radio 5 live.
When told there were images on social media of bottles being thrown towards it, he replied: "If you check the coach, there won't be any damage."
He also said their opponents should have arrived earlier.
"I don't understand why United couldn't get here at 4pm," he said. "They could have got here early. They knew it would be busy. It's crazy.
"There was congestion in the street and they couldn't get the coach in. There were people around the coach, but there was no attack on the coach."
Van Gaal claimed the Manchester United coach was only 100 yards behind the bus taking the West Ham players to the ground.
But the Dutchman refused to blame his side's defeat on the incident.
"It will have an influence, but I don't think we have to look for excuses," he said.
"We chased the match in the second half and 20 minutes before full-time we are ahead, so I don't think you have to look for that excuse."
A Metropolitan Police spokesman insisted that an "appropriate policing plan" was in place for the game, adding: "We are aware that a number of items were thrown towards Manchester United's coach."
In an update on Wednesday morning, the Metropolitan Police said four police officers had been injured during the trouble.
It added that no arrests have been made over criminal damage to the coach, but said three men were arrested during the game, including a 20-year-old on suspicion of affray and a 47-year-old and 18-year-old for pitch incursion.
Following the match, there were fireworks to mark the end of West Ham's 112-year stay at the Boleyn Ground.
A succession of taxi cabs brought a number of the club's former players onto the pitch, to join in the celebrations with the current squad.
But Jonathan Pearce, who was commentating on the game for the BBC, said the incident had "denied" many West Ham fans "a perfect evening".