The former prime minister has put his name to a report calling for tighter domestic controls and the negotiation of modified free movement rules with the EU.
He said the policy change would fulfil the will of the people expressed in last year's Brexit vote while allowing Britain to stay in the EU.
Mr Blair's intervention appears designed to provoke a fundamental shift in the Brexit debate and solve the seemingly intractable trade-off between the economy and immigration.
In an article for The Sunday Times's website, Mr Blair said: "There is no diversion possible from Brexit without addressing the grievances which gave rise to it.
"Paradoxically, we have to respect the referendum vote to change it.
"We can curtail the things that people feel are damaging about European immigration, both by domestic policy change and by agreeing change within Europe.
"This is precisely the territory the Labour Party should camp upon."
Mr Blair acknowledged that the approach marks a radical departure after he imposed no restrictions on immigration when eight countries from eastern Europe joined the EU in 2004.
But he said "back then the economy was strong, the workers needed", adding: "The times were different; the sentiment was different; and intelligent politics takes account of such change."
Mr Blair, an opponent of Brexit, went on: "If we go ahead with Brexit, we will have taken the unprecedented decision for a major country to relegate ourselves, like a top-six Premiership side deciding to play exclusively in the Championship.
"Other than President (Donald) Trump, I can't think of a single leader of any of our major allies or partners who thinks this decision is anything other than self-harming."