Mr Corbyn said he was carrying on with the responsibility given to him when elected leader by an overwhelming majority of members and called for the party to “come together”.
He swatted away claims that he half-heartedly campaigned for EU membership and so contributed to the vote for Brexit, saying that two thirds of Labour voters backed Remain.
The opposition leader said his party has won every byelection it stood in since he became leader and increased majorities in some seats. And he took credit for forcing the Tories into U-turns on cuts to tax credits, disability benefits and police funding, along with the forced academisation of schools.
His comments were certain to infuriate many Labour MPs who voted 172-40 to back a motion of no confidence in the leader.
Mr Corbyn is still in post a week after the vast majority of his shadow cabinet and frontbench team resigned in an effort to force him out.
In a defiant video posted on Twitter, Mr Corbyn called for unity. The Labour leader said: “After the events of the past week, I wanted to talk directly to Labour Party members.
“Only nine months ago I was very honoured to be elected leader of our party with 60% of the votes, I have a huge responsibility.”
He went on: “The membership has gone up by more than 60,000 in one week. We’re now at the biggest membership we’ve been, certainly in all of my lifetime.
“That membership wants and expects all of us — me as leader and members of parliament, to work together in their interests, the interests of everyone in this country, to achieve a better society, better standards of living and real equality in the future. That’s what the Labour Party stands for.”
He spoke after Angela Eagle delivered a clear threat to Mr Corbyn that she will mount a leadership challenge if he does not step down.
Ms Eagle, who was expected to launch her bid for the top job last week, warned that the threat of a contest had not gone away.