They believe the amendment could be disregarded because being in a customs union after Brexit is not British Government policy. Therefore ministers could simply state that no such steps had been taken.
A spokesman for the Department for Exiting the European Union said: "This amendment does not commit the UK to remaining in a customs union with the EU, it requires us to make a statement in Parliament explaining the steps we've taken.
"Our policy on this subject is very clear. We are leaving the customs union and will establish a new and ambitious customs arrangement with the EU while forging new trade relationships with our partners around the world."
It came after one Cabinet minister told MailOnline they would be able to "live with" the amendment on the grounds it would not actually stop Prime Minister May from pursuing her current Brexit policy.
The Government will now consider whether to try to remove the amendment from the Bill when it returns to the House of Commons.
It was the first of two defeats for the Government as peers also voted 314 to 217 in favour of an amendment to strengthen protection of employment rights and environmental standards post-Brexit.
Ministers have ruled out striking a customs union deal with the EU because it would restrict Britain's ability to do its own trade deals after Brexit.
Lord Forsyth, a Conservative former Cabinet minister, said the amendment was designed to "create division" in the House of Commons with the aim of "preventing Brexit going ahead".
He said: "What is going on here is an exercise by Remainers in the House who refuse to accept the verdict of the British people and I believe they are playing with fire."
Conservative Viscount Ridley said of the amendment: "It's an attempt to wreck this Bill and to prevent Brexit."
The amendment would force ministers to set out by the end of October this year what they had done to negotiate "an arrangement which enables the United Kingdom to continue participating in a customs union with the European Union".
The amendment was tabled by an alliance of Labour, Liberal Democrat and crossbench peers along with Tory rebels in a bid to soften the Prime Minister's approach.
They included the former Tory party chairman Lord Patten and former UK ambassador to the EU Lord Kerr - the diplomat who drafted Article 50, the legal mechanism for Brexit.
Lord Kerr said British negotiators should be able to "explore the idea" of a customs union as he claimed it would help provide a solution to the Irish border issue and protect trade.
He said that "other red lines" of the Government had already been "sensibly blurred" and that the customs union issue should follow suit.
The Lords will next consider the draft legislation on Monday, April 23.
