Sir Keir acknowledged that scrapping the cherished EU principle of free movement of labour would make it harder to maintain tariff-free access to the prized single market.
But he said the issue could not be ignored if the bitter divides caused by the EU referendum campaign are to be healed.
In a speech in London Sir Keir said: “No comprehensive approach to Brexit or response to the referendum result can ignore the issue of freedom of movement.”
Promising a “bold and ambitious” response, Sir Keir said the rules must change.
“The status quo is not an option,” he said. “Labour recognises that without the hard work and skill of migrants our public services, our businesses and our economy would suffer. But we have also always been the party that values strong, cohesive communities.
“It was striking that the referendum results showed the areas in the country with the highest levels of immigration voted most strongly to Remain. But the areas with the highest pace of change voted most strongly to Leave.
“That tells me that the British people are open and tolerant; but that they also expect change to be managed, rather than simply allowing the free market to rip through communities.”
But he acknowledged that demanding reforms to free movement rules would make securing single market access harder because the EU regards them as indivisible.
“This is not to pretend that arguing for changes to freedom of movement will not make a deal on single market access harder. It will.
“But in the negotiations to come, it is incumbent on the Government to fight for the fullest possible market access and reasonable management of migration.”
His comments came just days after shadow home secretary Diane Abbott, who speaks for Labour on immigration, said the UK could not “dump” free movement and stay in the single market.
She said: “Access to the single market and freedom of movement are inextricably linked, and it would be wrong - and the Labour Party has said this over and over again - it would be wrong to put the economy anything other than first.”
But Sir Keir said: “Our new relationship with the EU has to go beyond an economic argument and protecting our ability to trade in goods and services - vital though they are.”
If the Government loses its Supreme Court challenge and Prime Minister Theresa May is forced to get the backing of MPs to trigger Article 50, Sir Keir said Labour would seek to amend the legislation to prevent a “hard Brexit”.
He said: “Labour will push for a Brexit model which maintains our ability successfully to trade goods and deliver services with and to the EU.”
That would mean ensuring “continued tariff-free access” to the EU and avoiding extra bureaucratic burdens.
He insisted that there was no consensus among MPs for a hard Brexit and said Labour had already started consulting parliamentary experts about potential amendments.
But he denied that attempting to change the legislation would delay the Brexit process and prevent the Prime Minister from meeting her timetable of triggering Article 50 by the end of March 2017.