Mr Dowden, whose role involves responsiblity for party administration and overseeing the Conservative Campaign Headquarters, formerly Conservative Central Office, posted an early-morning statement on Twitter at 5.35am revealing the decision and publishing his resignation letter to the Prime Minister.
The contests, triggered by the resignation of disgraced Tories, offered voters the chance to give their verdict on the Prime Minister just weeks after 41 per cent of his own MPs cast their ballots against him.
Mr Dowden wrote: "Yesterday's Parliamentary by-elections are the latest in a run of very poor results for our party.
"Our supporters are distressed and disappointed by recent events and I share their feelings.
"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 said Conservative backbenchers and staffers "deserve better than this". He added that resignation was "a deeply personal decision I have taken alone".
In Tiverton and Honiton, Neil Parish – the Tory MP who had more than 60 per cent of the vote in 2019 – resigned after admitting he had watched pornography on his phone in the House of Commons.
A dramatic swing of almost 30% from the Tories to the Liberal Democrats saw Richard Foord secure a majority of 6,144.
The new Lib Dem MP used his acceptance speech to call for Mr Johnson “to go, and go now”, claiming his victory had “sent a shockwave through British politics”.
Mr Foord said: “The people of Tiverton and Honiton have spoken for Britain. They sent a loud and clear message – it’s time for Boris Johnson to go, and go now.”
He said that “every day Boris Johnson clings to office he brings further shame, chaos and neglect”.
In a message to the Prime Minister, Mr Foord, a former army major, said: “I can tell you that leadership means acting with decency and integrity.
“It means keeping your word. It means setting an example and putting other people’s needs before your own. I served alongside friends who personified these values, and laid down their lives in service of their country.
“And yet your behaviour Mr Johnson, makes a mockery of leadership. By any measure, you are unfit to lead. ”
The Liberal Democrats said Mr Foord had overturned the biggest majority in by-election history.
It was the sixth biggest swing against a government since 1945 in by-elections where both the seat and incumbent changed hands.
Lib Dem leader Sir Ed Davey said: “This should be a wake-up call for all those Conservative MPs propping up Boris Johnson.
“They cannot afford to ignore this result.”
He said it was “time for Conservative MPs to finally do the right thing and sack him”.
A Tory source said it was a “disappointing but not unexpected result” and “we are confident we will regain this seat at the next general election”.
In Wakefield, Simon Lightwood was elected with a majority of 4,925 on a swing of 12.7% from the Tories to Labour.
The previous Wakefield MP Imran Ahmad Khan quit after being found guilty of sexually assaulting a 15-year-old boy – a crime for which he was jailed for 18 months.
Wakefield was one of the so-called red wall seats won by the Tories in the 2019 general election after being Labour since the 1930s.
Mr Lightwood said: “The people of Wakefield have spoken on behalf of the British people.
“They have said, unreservedly: Boris Johnson, your contempt for this country is no longer tolerated.”
Labour leader Sir Keir Starmer said: “Wakefield has shown the country has lost confidence in the Tories.
“This result is a clear judgment on a Conservative Party that has run out of energy and ideas. Britain deserves better.”
He said the result showed Labour “is back on the side of working people, winning seats where we lost before, and ready for government”.
Mr Johnson, who is at a Commonwealth leaders’ summit in Rwanda, suggested it would be “crazy” for him to quit if the party lost the two seats and said mid-term by-elections were “never necessarily easy for any government”.