The force said it was called just after 17:00 BST on Sunday to reports of a woman being attacked by a dog in Commercial Road, Poplar, east London.
Footage on social media showed a man holding two dogs on nearby Limehouse Cut before he was Tasered and the dogs shot.
The Met said a man has been arrested.
One video showed a group of officers holding a catcher pole, riot shield and gun approaching the man and the dogs as he appears to walk away from them.
The police can be heard trying to persuade the man to surrender the dogs. The situation appears to become increasingly heated before the two dogs were shot.
In a statement, a Met spokesperson said: "Officers attended the location where the aggressive behaviour of two dogs was of considerable concern and posed a significant threat to them.
"A man was arrested in connection with the incident for having a dog dangerously out of control and assault offences. He has been taken into police custody."
The statement added a Taser was discharged during the incident and both dogs "were destroyed by police at the scene" but no-one was taken to hospital.
"This is never an easy decision for any officer to take, but police have a duty to act where necessary before any further injury is caused," the statement continued.
At the scene, a handwritten paper sign has been placed on a wall to mark the spot where the two dogs were killed, with a bunch of yellow flowers laid underneath.
One woman told BBC London reporter Matt Graveling they couldn't believe officers resorted to shooting the dogs.
"I don't think the dogs looked aggressive, they were both wiggling their tails," Jen said.
"It was crazy - there were people standing out on their balconies, it was five o'clock."
"The way of dealing with this with guns in the middle of the street, with us sitting on our balcony watching this, it just doesn't feel safe."
She said she disagreed with the Met's statement that the dogs posed a threat, saying it was "a choice they made" because, at that time, "the dogs were not aggressive".
Jen's partner Marcel said: "I was quite distressed, I was screaming at them, I was trying to stop it somehow but I couldn't."
He added: "We feel kind of powerless that stuff like this can happen.
"I guess obviously it's a scary situation being down here, but I'm not sure it was necessary to take something that looked like a gun and shoot the dogs."
The Met's directorate of professional standards reviewed the incident, including all of the available body-worn camera footage, and was "satisfied that there are no concerns around officer conduct", the force added.