Cabinet secretary David Lidington said the Government agreed with the "spirit" of the amendment, which is effectively "a statement of government policy" and said ministers would allow it to pass with a few tweaks to the legal language.
By a vote of 324 to 298, the House of Commons on Tuesday rejected a move to give lawmakers the power to send the government back to the negotiating table if they don't like the terms of the Brexit deal struck with the EU.
Theresa May faces a potential new front in the war between the pro and anti-EU wings of the Conservative Party after two days of votes that also exposed Labour's fault lines over Brexit. The Lords themselves would get chance to consider a motion, but that wouldn't mean a binding "yes" or "no" vote for them.
She told the Commons: "The central choice for Parliament is whenever we accept the outcome of the referendum or do we seek to subvert that process?"
As time ticks by, May can no longer kick decisions down the road, increasingly under pressure from European Union negotiators to come up with detailed positions not only on customs, but also on future trade relations and governance.
The pro-independence party accuses the British government of trying to seize powers that will be handed back from Brussels after Brexit and which the SNP believes should go to Scotland's Edinburgh-based parliament.
The British government needs to move on from internal debates and make choices about what it wants from Brexit, and must acknowledge that it can not have its cake and eat it, Irish Prime Minister Leo Varadkar said on Wednesday.
"We have now removed every incentive from the European Union for doing a deal by the end of November", one senior United Kingdom government official said. But how much of the proposed amendment has actually been accepted by the government?
But the resignation by Phillip Lee, who has always been critical of the government's Brexit strategy, underlined the deep rifts in the party over Brexit that makes such votes anything but easy.
Earlier, the Government managed to head off a major defeat on the EU (Withdrawal) Bill by offering a last-minute concession to Tory rebels that would give parliament a bigger say on the final deal.
With not all rebels persuaded that May's plan can prevent an economic shock after Brexit, some say they will challenge her plans to leave the customs union again during votes on other bills, on trade and customs, which will be brought back to the house some time before July 24.
But Brexit campaigners feared it could weaken Britain's negotiating stance in talks to leave the European Union and the Brexit ministry was quick to put out a statement saying: "We have not, and will not, agree to the House of Commons binding the government's hands in the negotiation".
It has also intensified pressure on a prime minister who lost her party's parliamentary majority at an ill-judged election previous year and tested her already weakened authority.
May, who leads a minority government propped up by the small Northern Irish Democratic Unionist Party (DUP), conceded that "we need parliamentary support" to implement Brexit.
It was not as simple on Tuesday, when May was forced to defuse another rebellion in parliament by offering a compromise that could hand lawmakers more control over Brexit.