Officers were called to reports of an altercation at the Intu Bromley Shopping Centre at 15:45 GMT.
Video filmed by a member of the public appears to show shoppers fleeing in panic, while a photo shows a police officer holding what appears to be a large knife.
A man in his 20s, who suffered a minor head injury, was arrested.
He was detained on suspicion of affray and possession of an offensive weapon.
Eyewitnesses reported seeing a man holding a machete.
Footage posted on Twitter by Alexander Adeosun, who was shopping with his friends at the time, shows people running through the shopping centre.
"There was panic and people were running and falling over," he said.
"My brother had to jump over the escalator."
Mr Adeosun said he was filming from the floor above and saw police arresting a man.
He said he also saw an officer placing what appeared to be a machete in a bag.
Keith Moon, 35, said he had been shopping with his wife at the time.
"We walked back in and saw the police leading a young guy in handcuffs away. One of the policemen was holding a machete around 2ft long," he said.
Armed officers were called to the scene but their assistance was not needed as it was not a firearms incident, the Met said.
A force spokeswoman said: "At this stage officers believe he sustained the injury as a result of an altercation.
Officers wish to trace "approximately six youths" seen running off from the scene, she added.
Earlier, the Met said it was seeking a man seen fleeing the scene, possibly discarding a weapon
Police recovered two knives from the scene.
A machete has been recovered after a suspected knife fight broke out at a shopping centre on Boxing Day.