Peers voted for the Tory amendment by 193 votes to 2 after a general debate on the White Paper on House of Lords reform which scraps hereditaries' voting and sitting rights.
Speaking during the debate, Lord Richard, the former Lords leader, stressed that the purpose of reform should be end the current "virtually unicameral system" by improving the second chamber's legitimacy.
The Labour peer, who initiated the reform process before he was replaced by Baroness Jay last summer, added: "Perhaps if Viscount Cranborne and I had been left alone, who knows what might have happened? There it is, we are all prisoners of our parties, to some extent."
Lord Cranborne was sacked as Opposition leader in the Lords by William Hague after he secretly negotiated a deal with Lord Weatherill, the chairman of the crossbenchers, to retain 91 hereditaries beyond stage one of reform.
Opening the debate, Lord Cranborne echoed Lord Richard's concern that Lord Wakeham should not accept the "by now notorious" passage in the Government's White Paper calling for significantly reduced powers for the Lords.
"If the House of Lords in its reformed state is to perform that essential and urgent service for the Commons, its composition must above all command universal authority and respect, and be independent," he said.
The Lords in its present form did not make more than "peripheral" amendments to Bills and stopped them from rejecting key legislation because they did not believe they had the standing or authority to do so "because of our composition".
He added: "We will need to use our existing powers to the full from time to time in a reformed House if we are to have any chance of frightening the Commons or holding the Government to account effectively."
Lord Cranborne accused ministers of having wanted until recently to stop reform after the voting and sitting rights of hereditaries were abolished.
"I suspect if you needed any more evidence of that fact, the rushed nature of the White Paper and the last-minute appointment of the Royal Commission membership are evidence enough to support that contention.
"They hoped that, when forced to reverse that contention during negotiations that took place in the latter part of last year, they could achieve the same objective by a fast stitch-up of a Royal Commission," he said.Reuse content