However, the House of Commons must approve the change before it can take force.
The cap would apply to borrowers who are unable to switch lender and whose mortgage is owned by an inactive or unregulated business.
The Lords voted 273 to 235 in favour of keeping the amendment, and the bill will now return to the House of Commons.
Martin Lewis, founder of MoneySavingExpert.com, said: “I am delighted that the Lords has seen the injustice that has been heaped on 100,000s of mortgage prisoners.
“While the government chose to bail out the banks in the financial crisis, it has never bailed out the banks’ customers who were victims of that collapse.
“Mortgage prisoners have been left paying obscene interest rates for over a decade, through no fault of their own.
“They have been completely trapped in their mortgages and unable to escape the financial misery this causes.
“Coupled with the devastating impact of the pandemic on people’s finances, the vote from the Lords is right to push for urgent action to prevent the situation from becoming catastrophic.
“The independent LSE report I funded has a cogent argument as to why an SVR cap isn’t a balanced long-term solution.
“But in lieu of anything else, I believe for those on closed-book mortgages it is a good stopgap while other detailed solutions are worked up – so this vote is an important move.
“And while it will be tough to get the Commons to enact it against government wishes, at the very least it ramps up the pressure on the government to come up with alternative solutions, at speed, which Rishi Sunak promised me, on the record, that he would do.”
