Footage on social media showed two protesters being bundled off stage within moments of unfurling their orange banners.
The group said on Twitter that its supporters were “demanding the UK Government immediately halt all new oil and gas consents and licences.”
“They attempted to address the audience before being forcibly removed”, the group wrote.
“Tonight’s action comes in response to the BBC’s underwhelming coverage of the climate emergency. In recent weeks, the BBC has been accused of ‘false balance’ as well as uncritically regurgitating government and oil company propaganda”, the group said.
The group said its members set off confetti cannons and sounded air horns – but this has been disputed.
A BBC Spokesperson said: “There was no disruption to the concert or the broadcast during the few seconds the protesters were on stage.”
Members of the audience can be heard booing during footage of the protest.
In response to the incident, Culture Secretary Lucy Frazer tweeted: “Eco zealots shouldn’t disrupt sports events, weddings or the Proms.
“My message is this: Leave people to enjoy the events they love, and stop damaging your own cause.”
The action is the latest effort by the climate protest group to disrupt major events this summer.
Protesters twice stopped play at the Wimbledon tennis championships when they ran onto onto a court last week.
That came a week after three people were arrested at Lord’s Cricket Ground on the first day of the second Ashes Test – including one man who was carried off the pitch by England wicketkeeper Jonny Bairstow.
Home Secretary Suella Braverman earlier this month held talks with senior sporting figures and police leaders on protecting major events from such action.