The government's welfare reforms took another battering on Monday night when a cross-party alliance, including many Liberal Democrat peers, voted to exclude child benefit from plans to impose a blanket £26,000 cap on household welfare benefits.

In the fifth significant government defeat on the bill, peers voted to temper the cap by 252 to 237.

Conservatives are convinced the cap, announced a year ago and due to come into force next year is hugely popular, and lambasted Labour for talking tough on welfare and then voting against the proposals. But the coalition assault was weakened by the number of senior Lib Dem peers, including Lord Ashdown and Lord Kirkwood, backing calls by bishops that child benefit should be excluded from the cap. Ashdown for the first time voted against the coalition government, intervening in the debate to ask what justification a minister could give morally or otherwise to withdraw a universal benefit for those on £26,000 but not for those earning £80,000.

The government will now have to decide whether simply to reinstate child benefit in the cap when the bill returns to the Commons, probably next week, or instead propose some clearer transitional relief. The mood in the government is to give little to no quarter. Figures such as Simon Hughes, the Lib Dem deputy leader, will be vital to the discussions.

An impact assessment published by the Department for Work and Pensions on Monday showed a total cap set at £26,000 a year would affect 67,000 households, including 90,000 adults and 220,000 children. The savings are worth £120m in a full year. Nearly 54% of the affected households are in Greater London, especially in inner city constituencies such as Hughes's in Bermondsey and Southwark. The removal of child benefit from the cap calculation reduces the number of households affected to 40,000.

A DWP spokesman said: "We are very disappointed by this decision and it clearly flies in the face of public opinion.

"There has to be a limit on the amount of money benefit claimants can receive. We think that limit is set at a fair rate of £26,000 – the equivalent to someone earning £35,000 before tax, a salary that many working families would be happy to receive.

"If you take child benefit out of the cap it will simply become ineffective, failing the very principles of our reforms, which is to bring fairness back into our welfare system while ensuring that support goes to those who need it. We are determined our reforms will be implemented in full and we will take this back to the House of Commons to reverse tonight's decision."

Lord Freud, the welfare minister, told peers the chief motive of the cap was not to reduce the size of the welfare bill but to change behaviour so incentives to work will increase. Labour countered that the motive could not be to end dependency since the benefits cap would be imposed on those on income support, and employment support allowance – two benefits that assume many of its claimants are not required to work.

Freud said the public saw the cap as being set at an appropriate level. He added: "One of the key drivers of family breakdown is unemployment that puts pressure on families. One of the most supportive things we can do for families is to make sure that work pays."

He argued that governments had paid out increasingly large sums with the result that some family members saw no advantage in going out to work, or looking after themselves, so depriving their family of positive role models.

"Like other welfare benefits, child benefit is provided by the state and funded by taxpayers and therefore we believe it is right it is taken into account along with other state benefits when applying the cap," he said.

"The effect of excluding child benefit would simply be that families on child benefit would have an income higher than average earnings – there would be no upper limit to the amount of benefit a household could receive as that would clearly depend on the number of children."

Labour's welfare spokesman Lord MacKenzie said he supported the cap, and the level set by the government. He added the cap will equally impact households where someone has just lost a job.

The government admitted it was "disappointed" by the result but insisted it intends to push through its plans "in full". A DWP spokesman said: "We are very disappointed by this decision and it clearly flies in the face of public opinion."

Enver Solomon, policy director at the Children's Society, said: "The Lords have stood up to the government and sent a clear message in support of children up and down the country. Children should not be held responsible and penalised for the employment circumstances of their parents.

"The government must not ignore the fact that the Lords have spoken out to defend the plight of some of the country's most disadvantaged children."

The Bishop of Ripon, John Packer, said: "It cannot be right for the cap to be the same for a childless couple as for a couple with children. Child benefit is the most appropriate way to right this unfairness." He argued that, in effect, the cap denied child benefit payments to people whose other benefits had reached £500 a week.

"This cap is not simply targeted at wealthy families living in large houses," he said. "It will damage those who have to pay high rents because often that rent has increased substantially in the course of their occupancy of that house."

Many argued the big welfare bill was a result of high rents in London.