The Welsh secretary says Labour MPs will have to look Leave-backing constituents “in the eye” and say they voted against their wishes if they reject Boris Johnson’s Brexit deal.
Alun Cairns said 80% of MPs were elected for parties pledging to honour the result of the 2016 referendum.
He said the deal could make Wales wealthier.
Mark Drakeford, Wales’ first minister, said Labour MPs will be “defending Welsh interests” in rejecting the deal.
He said he hoped the deal would be defeated when it comes to a vote on Saturday.
Prime Minister Boris Johnson has claimed he will win what is expected to be a knife-edge Commons vote on the deal on Saturday.
Speaking to BBC Wales Today, Mr Cairns said: “This is a time to act on the instructions where 80% of MPs stood on a manifesto to deliver Brexit.
“So I would say to those MPs, of all political persuasions that stood on a manifesto that said we would honour the outcome of the referendum, we will deliver Brexit – this is the time to do it.
“They will have to look at their constituents in the eye in the near future to say that they voted against their wishes, should they vote not to leave the European Union and against this deal.
“I think this deal answers the calls of the public, answers the needs of business and answers the excitement about the opportunity of leaving.”
Mr Cairns rejected claims the deal could lead Irish hauliers to avoid Welsh ports and take their goods direct to the EU mainland instead.
He said there is significant investment going on into the port of Holyhead, and there would be “extra opportunity of employment prospects in that part of world”.
He dismissed worries that the deal could lead to a weakening of workers rights. Provisions for the latter were moved from withdrawal agreement to the political declaration, which is not legally enforceable.
Mr Cairns said: “As we strike deals around the world, be that with the US, New Zealand, or Australia, there is no way we can compete on a low wage, low skill basis where rights are not as strong.”
But Mr Drakeford said the deal would damage Welsh companies by denying them access to the single market.
He said Labour MPs would be able to look their voters in the eye “because they will be defending Welsh interests. They will be standing up for Wales. It is time the secretary of state was willing to do the same”.
“I don’t believe that people who want to leave the European Union want to do it in a way that causes them, their families and their communities, real and lasting harm,” he said.
Pro-Brexit Monmouth Tory MP David Davies – who unlike Mr Cairns had campaigned to leave the EU – said Labour MPs who wanted to back the deal “just need to be honest with themselves”.
“Labour have got to make their minds up now,” he said.
“Are they against Brexit, in which case let them say so, let them be honest with the public and just say so, or are they going to vote for a deal?”
He added: “I would much rather leave with a deal than no deal – but they need to be in no doubt that we’ll be pushing hard for a no-deal Brexit if that’s the only way we can do it and respect the result.”
Pontypridd Labour MP Owen Smith, who was sacked as shadow Northern Ireland secretary by Jeremy Corbyn last year for calling for Labour to back another EU referendum, said any Labour MPs backing the deal would be “doing a disservice to their constituents”.
“For Wales it’s potentially really problematic – this notion of a hard border down the Irish Sea is obviously going to pose problems for Welsh ports, Holyhead and Pembroke, you can see significant difficulties,” Mr Smith added.
Former Labour Welsh and Northern Ireland Secretary Peter Hain said he feared for the “long-term stability” and peace process in Northern Ireland”.
He claimed the Brexit deal harboured “an agenda for a low-standards and low-tax haven – a Singapore-on-Thames”.
“Remember Brexit will not be finished with this deal. It’s just the beginning of years of uncertainty as Britain begins to negotiate new trade deals with the EU and 70 other countries.”
Plaid Cymru Westminster leader Liz Saville Roberts has led a cross-party letter to Chancellor Sajid Javid urging him to publish the impact assessments of the deal before MPs vote on the agreement on Saturday.
“If Boris Johnson is so confident that this deal is a good one, what good reason could he have for refusing to disclose vital details from the House of Commons?” she said.
With MPs voting this weekend on whether to back Mr Johnson’s deal on leaving the European Union, First Minister Mark Drakeford will visit Northern Ireland and the Irish Republic on Friday.
He will assure business leaders that Wales wants a strong relationship with firms whatever happens with Brexit.