Baroness Hollis's amendment, which passed 289 to 272, required a three-year "transitional period" with "full protection" for all families affected by the cuts.
Baroness Meacher's amendment, which passed 307 to 277, called for a government response to Institute for Fiscal Studies (IFS) analysis of the impact of the cuts to working tax credits. The IFS has said that the proposed cuts would cost low-income families an average of £1300 each year.
Downing Street criticised the votes last night, saying they raised a "constitutional issue".
"The Prime Minister is determined we will address this constitutional issue," a Number 10 spokesman said, adding, "A convention exists and it has been broken. He has asked for a rapid review to see how it can be put back in place."
Chancellor George Osborne said: "Unelected Labour and Liberal Lords have defeated a financial matter passed by the elected House of Commons, and David Cameron and I are clear that this raises constitutional issues that need to be dealt with."
He added: "However, it has happened, and now we must address the consequences of that. I said I would listen and that’s precisely that I intend to do. I believe we can achieve the same goal of reforming tax credits, saving the money we need to save to secure our economy, while at the same time helping the in the transition. That is what I intend to do at the Autumn Statement."
Earlier tonight, peers rejected a so-called fatal motion tabled by the Liberal Democrats, which would have killed the proposed cuts entirely, by 310 votes to 99. Labour peers abstained from the vote, in a move Lib Dem leader Tim Farron called "utterly depressing".
Farron said: “We support the delay in the proposals and the demand for transitional protection, but this alone won’t stop the Conservative’s attack on working families who rely on tax credits, or ensure that it really does pay more to be in work than remain on benefits."
"George Osborne must now go back to the drawing board and come back with plans to balance the books that don’t simply attack working families who are already struggling to get by," Farron added.
Shadow chancellor John McDonnell, meanwhile, called the votes a "huge blow" for the government.
McDonnell said: "The chancellor needs to understand that cutting on average £1,300 a year from over 3 million working families is not a sensible plan, and people are waking up to what Labour has been warning on this for months."
Repeating his earlier calls for a U-turn, McDonnell added: "George Osborne needs to now go away, and consider the only reasonable option open to him. If he U-turns fairly and in full on his tax credit cuts then I will support him on it, and so will the public."
Osborne told MPs on the Treasury Select Committee last week that he was "comfortable" with the tax credit cuts, saying: “This is fundamentally a judgement call, and I’m comfortable with the judgement call that I have made, and that the House of Commons has supported this week."
MPs have voted in favour of the tax credit cuts three times since Osborne first introduced the measures at the emergency summer budget in July.