"The time has come over Brexit, really, for economic reality and common sense to prevail over political dogma and wishful thinking", said Peter Mandelson, a member of the House of Lords from the main opposition Labour Party, who backed the amendment.
It would mean the government having to negotiate an worldwide agreement enabling Britain to continue to participate in the European Economic Area (EEA) after exit day.
Britain's upper house of parliament on Tuesday inflicted another embarrassing defeat on Prime Minister Theresa May's government on Tuesday, challenging her plan to leave the European Union's single market after Brexit. It was viewed as an attempt to pacify Brexit supporters who feared exit talks dragging on, but critics said it removed the flexibility to extend discussions if needed.
Voting for this cross-party amendment, spearheaded by the Conservative Duke of Wellington, was 311 to 233, majority 78.
The House of Lords, where the opposition Labour and Liberal Democrats outnumber PM Theresa May's Conservatives, has proven a consistent thorn in the government's side on Brexit.
She said: "On our future working relations with European Union agencies, many people, including within government, are only now becoming aware of the massive issues raised by our departure that ministers need to get right".
After the Lords, the bill will return to the House of Commons.
Labour's official stance was to abstain on the EEA amendment, but 83 of its peers rebelled against the frontbench.
Labour spokesperson Baroness Hayter urged peers to vote in favour of the amendment. The government opposed the change even though what it calls for is quite similar to what May has herself proposed.
And the opposite would kill off the federal government plan to write down the precise time and date of Brexit into regulation, which Mrs Could pledged final yr to stop insurgent MPs from derailing our exit.
The Government has suffered 14 defeats over its flagship EU Withdrawal Bill, with the latest reversals coming on the sixth and final day of the bill's report stage in the upper chamber.
"On laws it would mean the United Kingdom having to implement new European Union legislation over which in future we will have little influence and, of course, we will have no vote".
After completing its stormy passage through the House of Lords, the Brexit Bill will face its biggest test on its return to the House of Commons.