Speaking to broadcasters, Sir Keir said: “We need to get it right that’s why we’ve been talking to colleagues and having a constructive discussion.
“We’ve now arrived at a package that delivers on the principles with some adjustments and that’s the right reform and I’m really pleased now that we’re able to take this forward.”
As part of its climbdown, the government has reversed its plans to freeze the health-related component of universal credit. The payment will now rise in line with inflation for existing recipients.
Ministers will also carry out a review of the Pip assessment process, with input from disabled organisations.
The government originally hoped to save £5bn a year by 2030 with its Universal Credit and Personal Independence Payment Bill, aiming to slow the rise in claimants.
Working-age health-related benefits are estimated to cost an extra £30bn by 2029 without reforms.
But the government faced growing discontent from around 120 of its own MPs over the changes.
While the rebels told the BBC their colleagues are happy with the concessions, some Labour MPs have said they will still vote against the proposals.
The size of that rebellion is not yet clear, although it is thought some 50 Labour MPs could still vote against the bill, and there would be several abstentions.
It would take 83 Labour backbenchers – along with the other opposition parties – to defeat the government.