Theresa May has said the UK is "on course to deliver on Brexit" as she arrived in Brussels, the day after her first Commons defeat as prime minister.
She said she was "disappointed" at the vote on the EU Withdrawal Bill, but the legislation was making "good progress".
MPs backed an amendment giving them a legal guarantee of a vote on the final Brexit deal struck with Brussels.
EU leaders are expected to formally agree to start the next phase of negotiations on Friday.
The European Commission has said "sufficient progress" has been made on the first phase to move onto discussing the framework of a future relationship between the EU and UK - on issues such as security and trade.
Mrs May told fellow EU leaders at dinner on Thursday that she wanted to get agreement on the "implementation period" as a priority but wanted to talk about trade "as soon as possible".
She said was personally committed to delivering a "smooth Brexit" and she wanted to approach the next phase of talks "with ambition and creativity".
Speaking to reporters earlier, she said: "I'm disappointed with the amendment but actually the EU Withdrawal Bill is making good progress through the House of Commons and we are on course to deliver Brexit."
Asked whether she felt she would have to compromise more to win over rebels from her own party, she told BBC political editor Laura Kuenssberg: "We've actually had 36 votes on the EU Withdrawal Bill, and we've won 35 of those votes with an average majority of 22."
Mrs May lost by just four votes, as MPs backed an amendment to the EU Withdrawal Bill by 309 to 305.
Labour leader Jeremy Corbyn described it as a "humiliating loss of authority" for the PM and warned that his party would vote against another bit of the bill - the plan to put a fixed Brexit date into law.
He said setting an "arbitrary date" was not sensible and there "should be some flexibility".
What difference does this defeat make?
It will not derail Brexit but MPs who voted against the government hope it will give them a bigger say in the final deal Theresa May strikes with Brussels.
BBC Brussels reporter Adam Fleming said that as other EU leaders also run minority or coalition governments they would see the vote defeat as a small-scale domestic political issue.
The government had promised a "meaningful vote" for MPs on the final Brexit deal, but this defeat means that promise now has legal force and must happen before any UK-EU deal is implemented in the UK.
Ministers had wanted to be able to start implementing any deal as soon as it was agreed - in case, for instance, it was only agreed at the last minute.
BBC political editor Laura Kuenssberg said it would embolden the opposition and showed there was a majority in Parliament against a "hard Brexit".
Cabinet Minister Jeremy Hunt told the BBC the vote was "not going to stop Brexit".
How the government was defeated
Labour joined forces with the SNP, the Liberal Democrats and the Green Party in a cross-party alliance.
If all Conservative and DUP MPs had voted against the amendment the government would have won. But 11 Conservatives resisted the arm-twisting by their party managers to vote with the opposition.
The Tory rebels were Dominic Grieve, Heidi Allen, Ken Clarke, Jonathan Djanogly, Stephen Hammond, Sir Oliver Heald, Nicky Morgan, Bob Neill, Antoinette Sandbach, Anna Soubry and Sarah Wollaston. Another Conservative MP, John Stevenson, officially abstained by voting for and against the amendment.
Two Eurosceptic Labour MPs - Frank Field and Kate Hoey - voted with the Conservatives and the DUP.
The mood among Tory MPs
Emotions ran high before, during and after Wednesday's Commons debate, with Eurosceptic Conservatives accusing the rebels of trying to "frustrate" Brexit.
In dramatic scenes, the rebels shouted "too late" as Justice Minister Dominic Raab announced a concession shortly before voting began and Tory whips could be seen attempting to twist the arm of MPs thinking of voting against the government.
Leading rebel Anna Soubry said she had found a woman MP "upset and shaken" on Tuesday evening after a whip tried to persuade her not to revolt. She told MPs on Thursday morning, that none of the rebels took any pleasure in defeating the government, adding that "nobody drank champagne".
After the result was announced, one of the rebels, former cabinet minister Nicky Morgan, tweeted: "Tonight Parliament took control of the EU Withdrawal process."
This did not go down well with Tory MP Nadine Dorries, who called for the deselection of rebel Tories for "undermining the PM", and accused their leader, Dominic Grieve of "treachery".
Rebel Tory Sarah Wollaston hit back on Twitter, saying: "Get over yourself Nadine."
Tonight, the Tory rebels have put a spring in Labours step, given them a taste of winning, guaranteed the party a weekend of bad press, undermined the PM and devalued her impact in Brussels. They should be deselected and never allowed to stand as a Tory MP, ever again.— Nadine Dorries (@NadineDorries) December 13, 2017
This is not about a Govt defeat but about a Parliamentary victory Proud to have supported a #MeaningfulVote— Sarah Wollaston (@sarahwollaston) December 13, 2017
Dominic Grieve tried to calm the mood, insisting he was merely trying to ensure Brexit was carried out in an "orderly, sensible way".
BBC political editor Laura Kuenssberg
It's certainly true that the Tory party is so divided over how we leave the EU that the Parliamentary process was always going to be very, very choppy.
But another minister told me the defeat is "bad for Brexit" and was openly frustrated and worried about their colleagues' behaviour.
Back to Brussels for another summit
Theresa May has travelled to Brussels to attend a dinner with the 27 other EU leaders, at which she will urge them to approve an agreement to move Brexit talks on to a second phase.
They are all but certain to agree. Talks could then start next month on the two-year transition period the UK wants to ease it out of the EU after it formally leaves in March 2019.
But the EU wants more detail from the UK government before starting talks on a future relations - including trade - with the UK.
Brexit Secretary David Davis has said he wants to complete the "substantive portion" of trade negotiations by March 2019, leaving open the possibility that the detail will be hammered out during the two-year transition period.
The EU Withdrawal Bill - what is it?
The EU Withdrawal Bill is a key part of the government's exit strategy.
Its effects include ending the supremacy of EU law and copying existing EU law into UK law, so the same rules and regulations apply on Brexit day.
MPs have been making hundreds of attempts to change its wording - but Wednesday's vote was the first time one has succeeded.
Unless the government manages to overturn it further down the line, it means a new Act of Parliament will have to be passed before ministers can implement the withdrawal deal struck with Brussels.
The next Brexit row?
Labour's Shadow Brexit Secretary Sir Keir Starmer urged David Davis not to undermine Wednesday night's vote when the EU Withdrawal Bill reaches the next stage of its passage into law.
Mr Davis said the vote would lead to a "very compressed timetable" for Brexit legislation and the government "will have to think about how we respond to it".
There is also a row brewing over a vote next week on putting the precise date and time of Britain's exit from the EU - 11pm on 29 March 2019 - into law.
Sir Keir described the vote as the next "accident waiting to happen", telling Mr Davis: "Rather than repeat last night's debacle, will the government now commit to dropping that ill-conceived gimmick?"
Mr Davis told Sir Keir: "Unlike him, I do not view votes of this House of Commons as accidents. They are decisions taken by the House, and that decision we respect, as we will the next one."
Lib Dem Brexit spokesman Tom Brake also warned the prime minister she was heading for defeat if she did not drop the "silly idea" of enshrining the Brexit date and time in law, adding: "Parliament has now shown it is not prepared to be bullied."