The proposals, which would be the first legislation at EU level on the issue, still need to be agreed with the European Parliament before becoming law.
But EU lawmakers have similarly pushed for social media companies to do more to tackle hateful content on their platforms.
The proliferation of hate speech and fake news on social media has increased pressure on companies to remove such content promptly.
But internet campaigners have warned an excessive crackdown could endanger freedom of speech.
This week's agreement came a day after a suicide bomber killed at least 22 people at a concert in the English city of Manchester.
Ministers all offered their condolences to the British delegation.
Where the provision of videos forms an 'essential part' of the services provided by a social media company, they will have to take measures to block videos with hate speech, incitement to hatred and content justifying terrorism from their platforms.