The bill now goes to the House of Lords.
The UK Labour leader said on Sunday that being in "the single market is dependent on membership of the European Union".
But the unelected upper House of Lords may insist on further changes when the bill moves there for scrutiny later this month, while ministers still face anger from the devolved Scottish and Welsh administrations.
A further clause requiring the Government to carry out an assessment and prepare new legislation to prevent loss of environmental protections post-Brexit was defeated by an even slimmer margin of 17 - 318 votes to 301.
"This bill has never been fit for goal", said Labour's Brexit policy chief, Keir Starmer, describing any attempt to persuade the government that the legislation needed to change as "talking to a brick wall".
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'This is a critical piece of legislation that aims to maximise certainty for individuals and businesses after our exit.
The vast majority of Labour MPs voted against the bill tonight.
Justine Greening, who resigned as education secretary in a reshuffle earlier this month, warned that if Brexit did not work for young people, "it will not be sustainable" and they may seek to "improve or undo what we've done".
"The Government has not accepted any of the points that the opposition have made: they have conceded some ground on their own side - they have not taken seriously the propositions and the arguments that we have put forward on this side and that is unusual in my experience of dealing with Bills".
An analysis of the division list for the third reading vote showed four Labour MPs voted with the Government in support.
Greening's Conservative colleague, Dominic Grieve MP, warned ministers that leaving the European Union with no deal at the end of talks would be "The most catastrophic act perpetrated by a government in modern history".
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The bill was debated for around 80 hours in total, and effectively makes the British Parliament sovereign again.
The Welsh and Scottish governments have claimed they face a "power grab" from the United Kingdom government for proposing that Brussels authority in devolved matters like farming go to Westminster.
"It would replace current constraints on the National Assembly's legislative competence, which will fall away as a effect of the UK leaving the European Union, with a new set of constraints in devolved competences that would be controlled by the UK Government".
MPs from across the House used the debate prior to the vote to warn the government that they are prepared to stand in the way of a hard Brexit in future votes.
In their sights will be powers created to allow ministers to rewrite the UK's entire legal framework after Brexit with little scrutiny from parliament.
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They also predicted the measure would include a long-delayed renewal of a popular health insurance program for children. If that's the case, the burden will fall to House Speaker Paul Ryan who may have problems within his own ranks.