The tearful announcement was also greeted with warm tributes from the mutinous cabinet ministers who had helped force the prime minister into her premature departure.
Meanwhile, one leading Brexiteer made an immediate call for her successor to face down Brussels by demanding improved exit terms “to get Brexit done”.
The Labour leader turned on Ms May’s failure to tackle “burning injustices”, as she promised three years ago, claiming they were “even starker today”.
And he said: “The prime minister is right to have resigned. She has now accepted what the country has known for months: she cannot govern, and nor can her divided and disintegrating party.”
Mr Corbyn also repeated his call for a general election, rather than inflicting “yet another unelected prime minister”.
The two cabinet ministers who, a day before, had sealed Ms May’s fate by refusing to back her doomed withdrawal agreement bill were quick to praise her.
Jeremy Hunt, the foreign secretary, tweeted: “I want to pay tribute to the PM today. Delivering Brexit was always going to be a huge task, but one she met every day with courage & resolve.”
Sajid Javid, the home secretary, said: “Her dedication in taking our country forward has been monumental. She has served her country with fortitude and we are grateful to her for it.”
But David Mundell, the Scottish secretary, while saying Ms May had “my utmost respect”, added: “She has, however unfairly, become an impediment to the resolution of Brexit, and was no longer being given a hearing by parliament.
“Yesterday’s [European] elections will surely show that delivering Brexit is now more urgent than ever, and that will fall to a new prime minister. It's time to get on with the process of appointing one.”
Theresa Villiers, a leading Brexiteer sacked by the prime minister, made no reference to her record, while saying: “The new Conservative leader needs to bring the party back together and provide real leadership and direction.
“He or she should immediately seek improved exit terms from the EU. We need to get Brexit done and move on from the divisions it has caused in the party and the country.”
And David Mellor, a Cabinet minister under John Major, said he had “never been more embarrassed and humiliated” to be a member of the Conservative party.”
David Cameron also judged that Theresa May had “made the right decision” and praised her “strong and brave speech”.
“I know how painful it is to accept that your time is up and a new leader is required. She has made the right decision - and I hope that the spirit of compromise is continued,” he said.