Dame Cressida Dick said what happened to the 33-year-old “appals me” and made her “more determined, not less” to lead the organisation.
She welcomed the Home Secretary’s request for an independent investigation into the events – which she described as “fiendishly difficult policing”.
In ugly scenes on Saturday night, officers clashed with crowds who had gathered on Clapham Common to remember the marketing executive.
Dame Cressida said: “What happened to Sarah appals me. As you know, I’m the first woman commissioner of the Met, perhaps it appals me, in a way, even more because of that.
“What has happened makes me more determined, not less, to lead my organisation.
“I’ve listened to what people have been saying in the last week, I know that in the streets all across the UK women don’t feel as safe as we would all like women to feel. I am utterly determined.”
She said that “all the women and men of the Met are outraged at what has happened and they’re working as hard as they can to get justice for Sarah”.
“In that context, none of us would have wanted to see the scenes we saw at the end of yesterday’s events,” she added.
Earlier on Sunday, Priti Patel asked the Chief Inspector of Constabulary to conduct a “lessons learned” review into the events.
The Home Secretary spoke with Dame Cressida, having received her report into the police’s actions at the vigil.
But “in the interest of confidence in policing” Ms Patel asked Sir Thomas Winsor to conduct an independent review into what happened, a Government source told the PA news agency.
London mayor Sadiq Khan also said he would be asking HM Inspectorate of Constabulary and the Independent Office for Police Conduct to look into the events.
The mayor said the scenes at the vigil were “completely unacceptable” despite having received assurances from Scotland Yard last week that the vigil would be policed “sensitively”.
He said: “I asked the Commissioner and Deputy Commissioner to come into City Hall today to give me an explanation of yesterday’s events and the days leading up to them. I am not satisfied with the explanation they have provided.
“I will now be asking Her Majesty’s Inspectorate of Constabulary [HMIC] to conduct a full independent investigation of events yesterday evening and in previous days. I am also asking the Independent Office for Police Conduct [IOPC] to investigate the actions of police officers yesterday evening.
“It is vital that these events are not allowed to undermine the powerful calls since Sarah’s murder for meaningful action to finally stop men inflicting violence on women.
“It was clear before yesterday that there isn’t adequate trust and confidence from women and girls in the police and criminal justice system more widely. Further steps must now be taken to address this.”
Dame Cressida has faced calls to resign after the clashes in which her force’s officers were seen grabbing several women and leading them away in handcuffs.
The Metropolitan Police later said four people were arrested for public order and coronavirus regulation breaches.
Liberal Democrat leader Sir Ed Davey called on Dame Cressida to resign, while Women’s Equality Party co-founder Catherine Mayer said her position was “untenable”.
Labour leader Sir Keir Starmer said the Commissioner should not quit, but condemned the policing on Saturday as “wrong”.
He told reporters: “I was very disturbed to see the police action. I think it was wrong and I am pleased it is now going to be reviewed.”
Prime Minister Boris Johnson is said to have confidence in Dame Cressida and spoke with her on Sunday morning.
In one video obtained by the PA news agency, a woman could be seen being shoved forcefully in the back by two officers after being lifted from her knees.
The woman, who has not yet been identified, then tries to bend down near the officers and is shoved back again. She can be heard shouting that she is trying to retrieve her glasses.
Reclaim These Streets had organised the vigil before being forced to cancel following consultation with the Metropolitan Police, which said it would be in breach in coronavirus restrictions.
It has asked Dame Cressida for an urgent meeting so she can “explain the actions taken by the police last night, before she reports to the Home Secretary”.
After the clashes, organiser Jamie Klingler said the force’s handling of events was a sign of the “systemic ignoring and oppressing of women”.
The Fire Brigades Union added to criticism of the Metropolitan Police’s handling of the vigil, saying it was “shocking and unacceptable”.
“We utterly condemn the violence meted out by the Metropolitan Police last night on Clapham Common,” the union said on Twitter.
Martin Hewitt, chairman of the National Police Chiefs’ Council, said he would bring together police chiefs on Monday to discuss “what more we can do to better protect women”.
Hundreds of people converged on the south London park despite an official vigil being called off earlier in the day due to police warnings over coronavirus restrictions.
Vigils also took place in locations including Glasgow, Nottingham, Birmingham and Bristol.