Hammond said it was right to use monetary policy to support the UK economy during the "period of adjustment" as it seeks a new relationship with the European Union.
But his predecessor, George Osborne, said the rate cut - coupled with a £170bn package of quantitative easing, corporate debt purchases and incentives to lend - was only a "temporary solution" to the economic problems triggered by the June 23 referendum vote for Brexit.
And Labour's John McDonnell said it was down to the Chancellor himself to stimulate growth by reversing spending cuts scheduled by Mr Osborne for the years ahead.
Hammond said the Brexit vote created "a period of uncertainty, which will be followed by a period of adjustment as the shape of our new relationship with the EU becomes clear and the economy responds to that".
He added: "It's right that monetary policy is used to support the economy through this period of adjustment.
"That's why I have authorised the Governor's request for an increase in asset purchases and a new lending scheme to support the economy, helping ensure that the benefit of low interest rates is passed on by the banks to households and businesses.
"As recent figures on jobs and growth have shown, we enter this period of adjustment from a position of economic strength.
"And the Governor and I have the tools we need to support the economy as we begin this new chapter and address the challenges ahead."
But Mr McDonnell accused him of "dithering", on the day when Labour leader Jeremy Corbyn confirmed his plan for a £500bn boost to investment in infrastructure and industry.
"It's time for the Chancellor of the Exchequer to step up and shoulder his share of responsibility for economic stability," said the shadow chancellor.
"Britain is on hold waiting for Philip Hammond to tell us whether he will stick to his predecessor's planned cuts to investment, and firms and households can't wait until the autumn to find out...
"The more the Chancellor dithers, the greater the potential cost to the British economy."
Osborne - who before the referendum warned a Brexit vote was likely to result in interest rate rises - said the Bank was right to take action to support demand.
But he said the package was only a temporary answer which must be matched by permanent supply side reform - lower business taxes, free trade with the EU "and an unambiguous message we're open to overseas investment".
Labour leadership candidate Owen Smith urged the Government to implement "a radical fiscal policy to sit alongside interest rate cuts and greater QE".
Smith cited his own proposals for £200bn of investment over five years to rebalance and "reindustrialise" the economy.
Meanwhile, Liberal Democrat leader Tim Farron focused on the Bank's prediction of a rise to 5.5% in unemployment rates.
He described this as "a Brexit punch coming to households on our streets, putting jobs and livelihoods at risk".
"The Tory MPs and others who campaigned for Leave promised a new economic dawn, which we can clearly see is not going to break any time soon," said Mr Farron.
"It now looks like thousands of people will pay the price with their jobs for these spurious claims."
Labour MP Chuka Umunna, chairman of the Vote Leave Watch campaign, said: "The promises of Tory Leavers that Brexit would lead to higher growth and more jobs have never seemed so hollow.
"The myths told by senior figures in Theresa May's new Government will result in working people being worse off. They need to be held to account for the damage they have done to our economy."
The Institute of Directors questioned whether the rate cut would fuel business activity, pointing to a survey suggesting that six out of 10 of its members think it will have no discernible effect on their performance.
IoD economist James Sproule said: "The cost of capital is not the major factor for business at the moment.
"The Bank cannot do the heavy lifting on boosting business confidence. The Government has to play its part.
"The real test of confidence is going to be the Autumn Statement, which could be usefully brought forward several weeks."
Matt Whittaker, chief economist at the Resolution Foundation thinktank, said the Bank was "right to act with the limited armoury it has to restore some confidence in the economy".
However, he warned that it must be "poised to take further action if necessary".
Lee Hopley, chief economist at manufacturers' organisation EEF, said the Bank's action "should give confidence a lift", but the benefit of the rate cuts would be "limited" when they were already at such a low level.
"This puts the onus on the Government to take additional steps to support the UK manufacturing sector and economy," she added.
"Such measures need to form part of an industrial strategy that prioritises delivering a more competitive business environment for manufacturers."