The moves appear to break new ground for the social media platforms in dealing with the sensitive question of political disinformation.
Facebook, which has taken a hands-off policy on content from politicians, said this case merited fact-checking because it had been posted before the president shared it.
"Fact-checkers rated this video as partly false, so we are reducing its distribution and showing warning labels with more context for people who see it, try to share it, or already have," a Facebook spokesperson told AFP.
"As we announced last year, the same applies if a politician shares the video, if it was otherwise fact-checked when shared by others on Facebook."
Twitter used a new policy to identify misleading content for the first time.
The messaging platform - which is widely used by Trump - had announced it would start enforcing a policy against manipulated content, including from politicians, starting March 5.
The clip posted by White House social media director Dan Scavino appeared to show Biden telling a crowd: "We can only re-elect Donald Trump."
The footage was later retweeted by Trump and had been viewed nearly six million times by Monday. But the video cropped the end of Biden's sentence - made during a recent campaign speech Missouri state - where he was discussing the need for party unity in the wake of a bruising primary contest. "We can only re-elect Donald Trump if in fact we get engaged in this circular firing squad here," Biden had said.