Thousands of British jobs depend on frictionless trade with Europe, Greg Clark stressed as he said the controversial "customs partnership" plan was still on the table.
The Business Secretary also opened the door to extending a transition period on customs with the EU until a new arrangement was in place as Tory wrangling continued over how to replace the current system after Brexit.
Mr Clark said it could be a case of implementing a new customs arrangement "as soon as you can do", as he acknowledged a new regime may not be ready by the end of 2020 when the proposed transition period expires.
The Business Secretary was part of the Brexit "war cabinet" which failed to reach agreement on whether to back the hybrid customs partnership - which would see the UK collect import duties on behalf of the EU for goods arriving via British ports and airports - or the so-called "maximum facilitation" model relying on the extensive use of technology to minimise checks at the border.
Brexiteers have rejected the customs partnership, claiming it would keep the UK too closely tied to Brussels and Mrs May's inner war cabinet is thought to have lined up six-five against it, despite the Prime Minister and Mr Clark backing the model.
The Business Secretary denied being close to tears as he spoke at Wednesday's meeting but admitted "I do feel very strongly" about it, before highlighting the importance of having a customs deal with "the minimum of frictions" to firms operating a just-in-time production line.
He said Toyota, which employs 3,500 people at plants in North Wales and Derbyshire, was deciding where its next plant should be in Europe.
The nature of the business meant "50% of the parts come from out of the country", every hour, day and night and the customs partnership model means being able to "import them without any checks at the border, without any paperwork".
Mr Clark added: "You can compare models but actually you need to have in mind the future jobs but also the very important jobs for people today."
Asked if the transition could be extended until Britain was ready, he said: "It wouldn't be a question of extending the transition. It would be, as it were, implementing as soon as you can do ... there will be different parts that can be done immediately. There will be things that will take more time."
He was backed by former home secretary Amber Rudd, who said Mr Clark was right to argue the case "for a Brexit that protects existing jobs and future investment".
Business groups the British Chambers of Commerce and the CBI also welcomed Mr Clark's comments, saying it was important to maintain the status quo on frictionless trade until a new arrangement is in place.
But Brexiteers maintained their staunch opposition to the customs partnership.
In a sign of the divisions at the top of Government, a Cabinet source said: "Greg appears to be re-fighting the referendum, this is Project Fear 3.0.
"Instead of listening to vested interests he should pay more attention to the 17.4 million who voted to take back control of our trade policy.
"The Customs Partnership has been roundly rejected, making threats to keep it alive is just desperate."
Jacob Rees-Mogg, chairman of the influential European Research Group of Tory MPs hit out at Mr Clark's suggestion jobs could be at risk, saying: "This Project Fear has been so thoroughly discredited that you would have thought it would have come to an end by now."
He told ITV's Peston on Sunday: "The customs partnership is in a sense misnamed because it means single market as well as customs union and therefore we would not in effect be leaving the European Union."
Former Tory leader Iain Duncan Smith, writing on the ConservativeHome website, said: "Despite the crucial Cabinet sub-committee on Brexit negotiations and strategy deciding last week that the scheme isn't fit for purpose, some in Downing Street are, incredibly, now briefing out that after a few tweaks it can be presented again."
But pro-EU former business minister Anna Soubry urged Mrs May to "see off these ideologues who are blighting our party and blighting the Brexit process".
A Downing Street source dismissed as "nonsense" suggestions that Mr Clark's appearance and the support from pro-EU Tories and business organisations had been orchestrated from Number 10.
Writing in The Sun On Sunday, Mrs May said she had an "absolute determination to make a success of Brexit, by leaving the single market and customs union and building a new relationship with EU partners that takes back control of our borders, our laws and our money".
She said the UK was "making good progress towards that goal and we will carry on doing so with resolution in the months ahead".
DUP leader Arlene Foster, whose party props up Mrs May's government in Westminster, hit out at Brussels over the EU's stance on customs.
She told The Andrew Marr Show: "We don't believe that we have to stay in the customs union to have free flow between ourselves and the Republic of Ireland.
"We believe that there are ways to deal with this, and indeed, back in August of last year, as you know, the Government put forward various proposals. We were disappointed there wasn't the engagement from the European Union at that time."