Videos shared on social media show Metropolitan Police officers and stewards for the official event failing to stop the gatecrashers.
More than 100,000 people bought tickets for the 12-minute display set to music from Stormzy, Calvin Harris, Rihanna and Madness, with prices starting at £15 per person.
Held on the south bank of the Thames, next to the London Eye wheel, it was the first since 2019 and the first batch of passes sold out in a few minutes.
Some took their chances and stormed the barriers, running past police to get in.
A few officers can be seen trying to catch the gatecrashers, but their efforts were mostly unsuccessful.
The footage showed people tumbling to the ground in an attempt to evade them.
A statement from the force said: ‘From around 11.15pm until midnight there were a number of incidents where groups of people without tickets forced their way past hoardings and fencing.
‘Officers worked quickly in challenging circumstances to support stewards in an effort to prevent crowds getting into the ticketed area and to maintain the safety and security of those already inside.
‘At the junction of Storey’s Gate and Victoria Street, they had to draw their batons when a small but determined group who had pushed down fence panels in an attempt to get in became aggressive towards stewards and officers.
‘A small number of people were treated for minor injuries.
Over the course of the evening, eight people were arrested in the central London area.
These included assault on an emergency worker and possession of an offensive weapon.
A 31-year-old man arrested for sexual assault was wanted by the British Transport Police for five alleged indecent assaults against women.
He was arrested by officers during proactive patrols of the South Bank ahead of the New Year’s Eve event.
Two officers were also assaulted and sustained minor injuries.
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