
The Lib Dems overturned a huge Tory majority in Tiverton and Honiton, Devon, their third by-election victory over Boris Johnson’s party this year.
And Labour retook the seat of Wakefield, West Yorkshire, which it lost at the 2019 general election.
Mr Dowden said Tory supporters were “distressed and disappointed”.
In a letter to Mr Johnson, he wrote: “We cannot carry on with business as usual. Somebody must take responsibility and I have concluded that, in these circumstances, it would not be right for me to remain in office.”
Mr Dowden also called his resignation “a deeply personal decision that I have taken alone” following a “run of very poor results for our party”.
Speaking in Rwanda, where he is attending a Commonwealth heads of government meeting, Mr Johnson said: “We’ve got to recognise there is more we’ve got to do and we certainly will.
“We will keep going, addressing the concerns of people until we get through this patch.”
The by-election losses follow months of criticism of the prime minister over parties in Downing Street during lockdown.
They also come amid soaring inflation and concerns over the cost of living, and after a narrower-than-expected vote of confidence by Tory MPs in Mr Johnson earlier this month.
In Tiverton and Honiton, where former MP Neil Parish quit after he was found watching pornography in the House of Commons, the Lib Dems took 22,537 votes, beating the Conservatives by 6,144.
At the last general election, the Tories had a 24,239 majority in the mostly rural constituency.
In his victory speech, the Lib Dems’ new MP Richard Foord said: “Tonight, the people of Tiverton and Honiton have spoken for Britain.
“They’ve sent a loud and clear message: It’s time for Boris Johnson to go, and go now.”
Party leader Sir Ed Davey added: “The Liberal Democrats have made political history with this stunning win. It is the biggest by-election victory our country has ever seen.”
The Lib Dems also won by-elections in Chesham and Amersham and North Shropshire earlier this year, taking what had previously been safe Conservative seats.
A Conservative source called the result in Tiverton and Honiton “disappointing but not unexpected”, telling the BBC the party was confident it could retake the seat at the next general election.
Nothing reeks of panic quite like a resignation letter at 5:35am.
Not just any resignation letter. But the party chairman. Until now, at least, utterly loyal to Boris Johnson.
He manages five paragraphs, not one of which endorses him.
So, does he have confidence in Mr Johnson? He and his team are silent.
What we’ve seen overnight in miniature is the electoral pincer movement the Conservatives most fear.
Labour’s return to seats like Wakefield it could once nonchalantly bank.
The Liberal Democrats planting flags in England’s south west once again; a region that not long ago sent a minibus full of such MPs to Westminster.
Conservative MPs from the top down have the jitters this morning; the dawn decision of their former chairman quickening their pulse further.
In Wakefield, where a Labour victory had been largely expected, the party’s candidate, Simon Lightwood, won by 4,925 votes.
The previous MP, Imran Ahmad, resigned after being convicted of sexually assaulting a 15-year-old boy.
Reacting to his party’s win in Wakefield, Labour leader Sir Keir Starmer said it showed the “country has lost confidence in the Tories”.
“This result is a clear judgement on a Conservative Party that has run out of energy and ideas. Britain deserves better,” he added.
At the 2019 general election, Labour lost Wakefield – one of its so-called “red wall” of safe seats in northern England and the Midlands – for the first time since 1932.
Tory MP Simon Hoare, who has previously been critical of Mr Johnson, called Mr Dowden “an honourable man”, adding that he was “not responsible” for the by-election losses.