Theresa May has emerged unscathed from a packed meeting of the party’s backbench 1922 Committee after an “emotional and personal” speech reportedly won over MPs, despite doubts over her Brexit negotiating strategy.

The prime minister faced a handful of awkward questions from Brexiters including Nadine Dorries, Sir Edward Leigh and Philip Davies, but loyalists said she won over the room and there appeared little sign that a leadership challenge looked more likely.

Amber Rudd, the former home secretary, said that May had made an “emotional and personal” speech and won over colleagues, many of whom were angry with lurid anonymous briefings in the Sunday newspapers.

MPs present said Dorries demanded to know if May’s chief Brexit negotiator, Oliver Robbins, was freelancing when an extension to the post-Brexit transition period emerged last week, prompting the prime minister to defend the civil servant.

Leigh asked about preparations for a no-deal Brexit, observers said, while Davies pointedly asked May if the “17.4m who voted for Brexit” would get to see a deal they would recognise as representing a clean break from the European Union.

Most of the critical voices were familiar Brexiters.

May had volunteered to go to the meeting after intense speculation that backbench MPs would demand a vote of no confidence when Brexit talks with the European Union are a month or six weeks away from their conclusion.

Unnamed sources said at the weekend that May was entering “the killing zone”, and that if she turned up at the 1922 Committee meeting she had better “bring her own noose” – implying that she would face a critical reception at the meeting of her party’s MPs.

But the briefings appear to have backfired, prompting many party colleagues to voice support both before and at the meeting. It would require 48 MPs to write letters to Graham Brady, the chair of the 1922 Committee, to trigger a confidence vote. May would fall as leader if just over half the party’s 315 MPs were to vote against her, though anything apart from an emphatic victory could be fatal to her premiership.

One MP present said that Andrew Bridgen, a prominent Brexit critic who has denied being behind any of the briefings, demanded that May spell out “the three achievements that the government had chalked up in the negotiating process”.

Michael Fabricant emerged early from the meeting to a packed committee room corridor to tell journalists in a loud voice that the questioning of May was relatively tame. “It wasn’t Daniella and the lion’s den, it was a petting zoo,” he said.

Earlier, as she arrived in the committee room corridor, May smiled and joked with several MPs who could not squeeze into the room. “We support you anyway!” one shouted after her. Immediately after the meeting, May was due to see the Queen for a regular weekly audience at Buckingham Palace.

“She did seem relaxed, more so than I’ve seen her,” another MP said. “I think a lot of people have realised that slagging off your own side does not work. You step towards the edge of the cliff, consider the alternative, and you think, ‘Perhaps not.’”

Negotiations continue at official level in London and Brussels this week, with little immediate sign of a breakthrough over the two key stumbling blocks. May wants the EU to drop the Northern Ireland-only backstop, which would only apply if a future trade deal cannot be negotiated, and wants the EU to accept that her alternative UK-wide customs backstop is time-limited.

The UK is also discussing extending the transition period – which would see the UK stay in the single market and the customs union beyond 2020 – as an alternative to relying on any backstop, although that could require Britain to pay billions more in contributions to the EU budget.

May faced fresh questions about that plan after the European parliament was informed that it was the British prime minister who first raised the possibility of extending the transition period beyond 2020 with the EU’s leaders. Donald Tusk, the president of the European council, told MEPs it was May who brought up the idea at last week’s summit, despite the prime minister’s public claims that she is vehemently opposed to such a move.

Tusk said: “Since Prime Minister May mentioned the idea of extending the transition period, let me repeat that if the UK decided that such an extension would be helpful to reach a deal, I am sure that the leaders would be ready to consider it positively.”

Under the terms of the 21-month transition period so far negotiated, the UK would continue to live under EU rules and laws, but without any say in their design, as the country would not have any representation in the bloc’s institutions.

The former Ukip leader Nigel Farage, who remains an MEP, immediately seized on Tusk’s comments. “Mr Tusk, I want to thank you for confirming that it was Theresa May that asked for the one-year extension to the transition period,” he said.

• This article was amended on 25 October 2018 to remove comments mistakenly attributed to Helen Whately, party vice-chair.