New clause three, proposed by Labour leader Jeremy Corbyn, was narrowly defeated by 333 votes to 284, a majority of 49.
The Opposition argued such an amendment to the European Union (Notification of Withdrawal) Bill was needed to ensure parliamentary oversight of the EU withdrawal negotiations.
Brexit minister David Jones said the Government rejected the amendments on scrutiny, adding: "A lot of them are unnecessary because what they're seeking to achieve is effectively already being done by the Government.
"No-one can deny that the Secretary of State has been absolutely assiduous in his engagement with Parliament.
"It's been the source of intense scrutiny over the past seven months."
MPs have rejected a Labour amendment to the Brexit bill
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Remainer MPs have tabled hundreds of amendments to the Bill, with votes on proposed clauses set to carry on until Wednesday.
But Mr Jones said many of the amendments had "virtually nothing to do with the Bill".
On the status of EU nationals, he said: "This is less an issue of principle than one of timing, with a few EU countries insisting, frankly, that there can be no negotiation without notification and that therefore nothing can be settled until Article 50 is triggered.
"We could not be clearer about our determination to resolve this issue at the earliest possible opportunity, ensuring the status of UK nationals in the EU is similarly protected.
"Some have called for a unilateral guarantee now, but we have a very clear duty to UK citizens living in other member states, of whom there are about one million, to look after their interests and provide as much certainty for their futures as well.
"The suggestion from some, effectively that we should offer that unilateral guarantee to EU nationals in the UK, whilst at the same time failing to achieve security for our own nationals abroad, is a course that would carry the risk of a prolonged period of stressful uncertainty for them, which we are not prepared to accept."
Theresa May said she wanted to get on with leaving the EU
Theresa May had earlier told the Commons to "get on with it" and pass the bill that gives her the power to trigger Article 50 and begin the process of withdrawing from the EU.
She said: "Our European partners now want to get on with the negotiations, so do I, and so does this House, which last week voted by a majority of 384 in support of the Government triggering Article 50.
"There are of course further stages for the Bill in committee and in the Lords and it is right that this process should be completed properly.
"But the message is clear to all – this House has spoken and now is not the time to obstruct the democratically expressed wishes of the British people.
"It is time to get on with leaving the European Union and building an independent, self-governing, global Britain."
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