Stephen Barclay also said Theresa May could stay in post if, as expected, MPs reject her Brexit plan.
The PM has warned Tory rebels it could lead to a general election, and there was a "very real risk of no Brexit".
Boris Johnson said the PM could stay on if she lost but must renegotiate the deal with Brussels.
The withdrawal deal negotiated between the UK and EU has been endorsed by EU leaders but must also be backed by Parliament.
But the government is widely expected to lose the vote with Labour, the Liberal Democrats, the DUP, the SNP and dozens of Conservative MPs saying they cannot support the deal.
European Council President Donald Tusk tweeted on Sunday it would be "an important week for the fate of Brexit", after a phone call with Theresa May.
Despite newspaper speculation, it could be called off, Brexit Secretary Stephen Barclay said: "The vote is going ahead and that's because it's a good deal and it's the only deal".
He warned against suggesting negotiations could be reopened: "The French, the Spanish and others will turn round, if we seek to reopen the negotiation, and ask for more."
And he insisted that Theresa May could remain, prime minister, even if her deal is heavily defeated by MPs: "Yes, the prime minister is fighting for us and will continue in the post."
What did Boris Johnson propose?
Mr Johnson, who quit the cabinet over Mrs May's Brexit strategy, told the BBC he did not want a "no-deal" Brexit or another referendum, but it was not right to say there were no other alternatives.
He said the Northern Ireland "backstop" - effectively an insurance policy to prevent a hard border if no free trade deal is struck in time - put the UK in a "diabolical negotiating position".
MPs could give Mrs May "a powerful mandate to change that backstop" by voting it down on Tuesday, he said.
The border issue should instead be covered in negotiations over a future trading relationship with Europe, he said, and the UK should "incentivise" the EU by withholding "at least half" of the UK's agreed £39bn "divorce" payment, until a free trade is signed at the end of 2020.
He dismissed suggestions he had already offered other Tory MPs jobs if they backed a future leadership bid by him as "nonsense" and said the public wanted to see a plan "to get out of this mess" rather than "stuff about leadership elections and personalities" - although he did not rule out challenging Mrs May for the leadership.
And the former foreign secretary, a key figure in the 2016 campaign to leave the EU, said: "Don't underestimate the deep sense of personal responsibility I feel for Brexit and what's happened.
"It breaks my heart to think that after all we fought for... that we should consign ourselves to a future in which the EU effectively rules us in many respects and yet we have no say around the table in Brussels."
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