Many who spoke in the debate vote said they had received letters from people worried about the effects on their families
The Government has been defeated over its Brexit bill in the Lords, with peers demanding protection for EU citizens in the UK.
A total of 358 voted for a Labour amendment to the bill to trigger Article 50, with 256 voting against.
It called on ministers to, within three months of Article 50 being triggered, unilaterally bring forward guarantees for the 3.5 million people from the bloc living in Britain.
In an attempt to head off the defeat, Amber Rudd had written a letter assuring peers that EU citizens would be treated with the utmost respect.
Labour, Lib Dem and some crossbench peers had said they felt a unilateral statement of support was necessary to remove the insecurity many felt ahead of Brexit.
Many who spoke in the debate preceding the vote said they had received many letters from people worried about the effects on their families.
Earlier, the Government's spokesman on Brexit in the Lords, Lord Bridges of Headley, had made an impassioned plea for the amendment to be defeated.
He said: "The status of EU nationals is one of the most emotive issues created by this debate. The lives of over four million people who have chosen to make this country their homes.
"I'd like to make two core points. First... between now and the date of exit, nothing will change for EU nationals living in the UK.
"Any EU citizen who is exercising EU treaty rights or has the right to remain, continues to have those rights to remain. There is no need to apply for a document to prove this.
"An EU national who has been here for five years, does not have to apply for a document to prove their right to be here permanently.
"The bill to repeal the European Communities Act (the Great Repeal Bill) will not be used to change our immigration system. This will be done through a separate immigration bill.
"After we have left the EU, the UK will continue to be bound by and observe the ECHR (European Court of Human Rights), which, in accordance with Article Eight, and appropriate case law, will protect EU nationals' rights to ... their private and family life.
"So... nothing will change for any EU citizen... without this parliament's approval."
He went on to remind the house that the issue of the long-term rights of EU citizens in the UK and UK citizens throughout the bloc will be dealt with early in the forthcoming Article 50 negotiations.
The sponsor of the amendment, shadow Brexit minister Baroness Hayter of Kentish Town, had said she was concerned EU nationals would be used as "bargaining chips" in negotiations to quit the EU.
Lady Hayter said EU nationals here and British expats living in Europe should not be "traded against each other".
Several Conservatives, including Viscount Hailsham, backed the amendment in the Lords, which has no government majority.
Others, like Tory former chancellor Lord Lawson of Blaby, shared the concerns but said the amendment had no place in the bill as it stood.
Government sources said ministers will now seek to overturn defeat in the Commons.
If they succeed it would then go back to the Lords, although analysts say it is unlikely peers would force a second vote.
However, the process could delay the bill.
Theresa May is understood to be planning to trigger Article 50 around 15 March and has long had a deadline of the end of March.
(Additional reporting from IRN)