But Mrs May accused Mr Johnson of "tearing up our guarantee to the people of Northern Ireland" that there would be no hard Irish border, arguing that a Canada-style deal could not apply to the whole UK.
In the face of Labour's appeal to those suffering from the effects of austerity, Mrs May said Tories must "defend free markets, because it is ordinary working people who benefit".
In her crucial keynote speech to the Conservative conference in Birmingham, Mrs May said next year's post-Brexit Spending Review will set out a programme of increased investment in public services, as a mark that the decade of cuts following the financial crash is coming to an end.
The pound dropped 0.4% on the day to as low as $1.2963, close to yesterday's three-week low of $1.2941, before recovering to $1.2990 later.
Foster was challenged on her opposition to equal marriage on Tuesday by left-wing columnist Owen Jones, during an interview at the Conservative Party conference in Birmingham.
In her speech, May again stuck to her Chequers plan, named after her country residence where she hashed out an agreement over the proposals in July.
"The new skills-based system will make sure low-skilled immigration is brought down and set the United Kingdom on the path to reduce immigration to sustainable levels, as we promised", May was quoted as saying.
Foster went on to say that she believes people are selling Prime Minister Theresa May short in her capacity to get the deal done. This was despite the Tory leader's team meticulously putting together a media plan to keep off the front pages.
Johnson did not call on Tuesday for May to be replaced, saying she should simply change course.
She said: 'It would be tough at first, but the resilience and ingenuity of the British people would see us through'.
To be fair: May is entitled to feel exasperation with the hand that she has been dealt, with her toxically fractious party and with the belligerent Brussels EU leadership, but when everything is said and done she knew what she was getting herself into.
May and her team face weeks of hard conversations with Brussels to win a deal, but she also faces challenges from inside her own party and from her partners in parliament, Northern Ireland's Democratic Unionist Party. "We will never undermine the principles of the European project to rescue a political party in Britain who is not even capable to find a common line on the exit of the mess Brexit". Taking back control of our borders, laws and money.
Labour Leader Jeremy Corbyn, she said, would "outsource our conscience to the Kremlin".
It also appeared aimed at showing the prime minister to be at ease and in charge at a momentous moment for Britain.
To applause from activists, she said that "sound finances" would remain an essential part of the Government's economic policy, but would no longer be "the limit of our ambition". "I wanted to be convinced", said Carol Ward. May out as leader, convinced there is no clear alternative and that Mr. Johnson isn't up to the job.