Keir Starmer, Labour's chief Brexit spokesman, said the proposal was "a significant climbdown from a weak government on the verge of defeat".
The government won the first five votes late yesterday on the EU (Withdrawal) Bill, which seeks to formally end Britain's membership and transfer European law onto the United Kingdom statute books.
Media captionWhat's going on with the EU Withdrawal Bill?
A new clause tabled by Labour former minister Chris Leslie, which would require ministers to produce a report on how European Union law will be applied during a transition period, was defeated by 316 votes to 296.
More than 400 amendments have been tabled to the EU Withdrawal Bill, which is created to convert EU law into United Kingdom legislation by the end of March 2019.
However, Brexit secretary David Davis conceded that even if MPs failed to back that legislation - the withdrawal agreement and implementation bill, Britain would still leave the European Union on March 29, 2019.
Ministers want to set exit day as 2300 GMT on March 29, 2019.
The MPs later won support from Labour colleagues fighting against a hard Brexit with Chuka Umuna, the former shadow cabinet minister, saying the front page was "an ideological witch-hunt to bully Tory MPs to go against their instincts to do what they think best for our country".
One of them, former attorney general Dominic Grieve, said it was "utterly pointless and counterproductive" and would remove any flexibility in case the negotiations were delayed.
It comes despite 56 per cent of Stafford constituents - some 43,386 people - voting to leave the EU.
Asked if it was MPs' "duty to scrutinize that legislation, debate and consider amendments", the Prime Minister replied that was "right".
The EU (Withdrawal) Bill is a key part of the government's strategy for leaving the EU following last year's referendum.
Both Scotland's and Wales' devolved governments have expressed fears the "power grab" legislation will return responsibilities from Brussels to London, rather than to their countries' administrations.
The government said it wants an implementation period of around two years after Brexit to stop an economically damaging "cliff-edge" - but insists Britain will be fully out of the EU.
"Discussions will continue and hopefully we can reach some points of agreement in the weeks to come".
The amendment fixing the date of Brexit will not be voted on until next month at the earliest, and the Government survived the early skirmishes in the battle to get the so-called repeal bill through the Commons.