The CEO's statement follows allegations that 50 million Facebook users' private information was misused by the political consultancy firm.
Mr Zuckerberg pledged to introduce a series of changes.
These would make it far harder for apps to "harvest" user information, he said.
A breach of trust between app creator Aleksandr Kogan, Cambridge Analytica and Facebook had occurred, Mr Zuckerberg said in a statement on his Facebook page - his first public comments since the scandal broke.
But he added it was also a breach of trust "between Facebook and the people who share their data with us".
He continued: "We have a responsibility to protect your data, and if we can't then we don't deserve to serve you.
"I started Facebook, and at the end of the day I'm responsible for what happens on our platform."
Earlier Mr Kogan, the Cambridge University academic who created the app that harvested data from 50 million Facebook users - mostly in the US - said Cambridge Analytica and the social media firm have made him a "scapegoat".
He insisted he did not know his work for Cambridge Analytica in 2014 violated Facebook's policies.
To address current and past problems, Mr Zuckerberg said Facebook would:
- investigate all apps that had access to large amounts of information before the platform was changed "to dramatically reduce data access" in 2014
- conduct a full audit of any app with suspicious activity
- ban any developer that did not agree to a thorough audit
- ban developers that had misused personally identifiable information, and "tell everyone affected by those apps"
In future, he said Facebook would:
- restrict developers' data access "even further" to prevent other kinds of abuse
- remove developers' access to a user's data if the user hadn't activated the developer's app for three months
- reduce the data that users give an app when they sign in to just name, profile photo, and email address
- require developers to obtain approval and also sign a contract in order to ask anyone for access to their posts or other private data
Mr Zuckerberg added: "While this specific issue involving Cambridge Analytica should no longer happen with new apps today, that doesn't change what happened in the past.
"We will learn from this experience to secure our platform further and make our community safer for everyone going forward."