The age limit had been backed by peers earlier this year after growing calls from campaigners, including actor Hugh Grant.
Supporters of the Australian-style ban said parents are in "an impossible position" over the online harms their children are being exposed to.
Others, including the National Society for the Prevention of Cruelty to Children, warned a ban could drive teenagers into unregulated corners of the internet.
MPs voted 307 to 173, a majority 134 on Monday, against the proposed change to the Children's Wellbeing and Schools Bill, which was brought forward by Conservative former minister John Nash.
However, a ban could still come in future after the Commons supported a government bid to give additional powers to the secretary of state.
Under the amendment in lieu, Science Secretary Liz Kendall could "restrict or ban children of certain ages from accessing social media services and chat bots".
She could also limit children's VPN use, restrict access to addictive features, and change the age of digital consent in the UK, Education Minister Olivia Bailey told MPs.
"Many parents and campaign groups have called for an outright ban on social media for under-16s," Bailey said.
"Others, including children's charities, have warned that a blanket ban could drive children towards less regulated corners of the internet or leave teenagers unprepared when they do come online.
"That is why, last week, the government launched a consultation to seek views to help shape our next steps and ensure children can grow up with a safer, healthier and more enriching relationship with the online world."
The consultation will look at whether social media platforms should come with a minimum age requirement and whether platforms should switch off addictive features such as autoplay.
Meanwhile, 107 government MPs abstained from the vote, including Sadik Al-Hassan.
"Parents like me are locked in a daily battle that they simply cannot win alone, fighting platforms that have been specifically designed to keep children hooked," he said.
"If a drug were causing such measurable harm for 78 per cent, it would be withdrawn, reformulated or placed behind a counter with strict controls on who could access it.
"We have an identifiable source, we have overwhelming evidence of harm, and we have the power to act."
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