Addressing what is known as the Conservative's 1922 committee, May told the politicians: "I got us into this mess, and I'm going to get us out of it."
As she spoke at the private meeting, party managers continued talks with the Democratic Unionist Party (DUP) of Northern Ireland about its 10 MPs working with her 318 Conservative group of MPs to give them a majority in the House of Commons.
MPs from all parties are due to gather at Westminster Tuesday to vote on whether John Bercow should remain as Speaker of the House of Commons.
That had been scheduled to lead to a state opening of the new session of parliament next month when Queen Elizabeth II outlines the proposals May's government intend to put forward.
Discussions with parliamentary officials continued Monday night to decide whether the state opening ceremony should be delayed by several days.
Talks are due to start next Monday between government ministers and the European Union to start formal negotiations for Britain's departure from the EU.
Those talks in Brussels could also be delayed, but the government insist the two-year Brexit timetable would still be adhered to.
A leading Conservative MP who was present at the 1922 committee meeting told the Guardian newspaper later that May was "contrite and genuine but not on her knees" as she apologised to MPs for the election result that cost the Conservatives its majority in the House of Commons.
May is also said to have expressed her apologies to the MPs who had lost their seats in the snap election which she had called.
Media reports said that May had also assured MPs that the working arrangement with the DUP would have no impact on issues such as gay rights.
Political commentators said later that May had steered herself through the first meeting of her cabinet at 10 Downing Street, and then through the meeting of backbenchers, easing at least for the moment calls for her to be replaced as prime minister.
Foreign Secretary Boris Johnson gave his take of May's performance at the 1922 committee. He wrote on his social media site: "Stonking performance by the PM at 1922. One team going forward together for the UK."
Conservative veteran Nicholas Soames said after the meeting: "Unreserved support for PM at 1922. Struck just the right tone and dealt with all substantial points. Party united behind her."
May is due to hold a meeting at Downing Street Tuesday with DUP leader Arlene Foster to discuss the working relationship between the two parties.
On the sidelines, the main opposition Labour Party plans to contest the Queen's speech in the hope that leader Jeremy Corbyn will move into 10 Downing Street as the new prime minister.
