In a heavy defeat for the Labour government on Wednesday, peers supported the cross-party move by 261 votes to 150, majority 111, meaning it will have to be considered by MPs in the Commons.
Supporters of the Australian-style ban have argued parents are in "an impossible position" with regard to the online harms their children are being exposed to.
Technology Secretary Liz Kendall announced a three-month consultation this week, which will consider the advantages and disadvantages of a ban, as well as possible overnight curfews and actions to prevent "doom-scrolling", before reporting back in the northern summer.
However, Tory former schools minister Lord Nash, who spearheaded calls for a ban, argued the late concession simply represented more delay.
"The government's consultation is, in my view, unnecessary, misconceived and clearly a last-minute attempt to kick this can down the road," he said.
"The evidence is now overwhelming as to the damage that this is causing," he told parliament.
"There is now so much evidence from across the world that it is clear that by every measure, health, cognitive ability, educational attainment, crime, economic productivity, children are being harmed.
"This vote begins the process of stopping the catastrophic harm that social media is inflicting on a generation," he said after the vote.
Hollywood star Hugh Grant was one of the signatories in a letter to Prime Minister Keir Starmer, Conservative leader Kemi Badenoch and Liberal Democrat leader Sir Ed Davey, backing the amendment.
Other signatories included singer Peter Andre and actress Sophie Winkleman, while the letter was also backed by Parentkind and Mumsnet.
"Children are being served up extreme content without seeking it out. Parents know this has to stop. But they cannot do this alone, and they are asking for politicians to help," the letter said.
Peers also backed by 207 votes to 159, majority 48, a ban on providing VPN services to children over concerns they can be used to bypass age verification restrictions on accessing adult content.
Changes made by peers to the Bill will be considered by MPs during the process known as ping-pong, when legislation is batted between the Commons and Lords until agreement is reached.
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