But despite jubilation among Mr Corbyn's supporters at bloodying Mrs May' s nose, Labour MP Chris Leslie said the party should not pretend it achieved a "famous victory".
May is under pressure after the Conservatives lost their parliamentary majority in Thursday's election.
May called the early election in April, when opinion polls suggested she was set for a sweeping win.
British Prime Minister Theresa May is now fighting for survival after a failed election gamble undermined her authority and plunged the country into a major political crisis days before the start of talks to leave the European Union.
Downing St. chiefs of staff Nick Timothy and Fiona Hill, who resigned Saturday, formed part of May's small inner circle and were blamed by many Conservatives for the party's lacklustre campaign and unpopular election platform, which alienated older voters with its plan to take away a winter fuel allowance and make them pay more for long-term care.
Timothy, who has worked with May since she was in the Home Office, hoped MPs would get behind the Prime Minister for the good of Brexit. By Saturday, there were reports that she might even be ousted as prime minister by her party.
The agreement came after Theresa May sent her Chief Whip Gavin Williamson to Belfast to negotiate with the Northern Irish party. "May stares into the abyss", wrote The Times, while the Conservative-supporting Sun tabloid said succinctly: "She's had her chips".
The spokesman indicated this would not be a formal coalition but a minority government with looser DUP support on a "confidence and supply basis".
And in a phone call with the German Chancellor Angela Merkel, the PM attempted to show it's business as usual by confirming she'll start Brexit talks as planned in the "next couple of weeks".
United Kingdom election: 51 black, Asian, ethnic minority MPs elected to parliament
She ignored the anxiety that Brexit has created and the economic consequences that are now just beginning to bite. Mrs May is seeking a deal with the Democratic Unionist Party after losing her majority in Thursday's election.
EU Budget Commissioner Guenther Oettinger said it may now be possible to discuss closer ties between Britain and the EU than May had initially planned, given her election flop. She also reiterated that she would seek a reciprocal agreement early in the talks on rights of European Union and British citizens, Downing Street said.
"For instance, if London were to stay in the customs union, then it would not have to renegotiate all trade agreements", he told the Frankfurter Allgemeine Sonntagszeitung newspaper.
The pound on Friday fell 1.7 per cent against the USA dollar and 1.4 per cent against the euro.
She insisted she would press ahead with Brexit talks, which are to begin in 10 days.
Pressed if he was being clear that the United Kingdom will leave the EU, Mr Corbyn said: "Absolutely".
Fallon also said he believed there was a majority in parliament for this Brexit plan.
Party insiders are placing bets on how long May will last, less than a year after Britain's surprise referendum decision propelled her into Downing Street. "She needs to be more collegiate, seeking the advice of the cabinet. while reducing her nearly total reliance on a tiny cadre of advisers".
"Imagine she survives until autumn of next year", he said.
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