The ban - which will not take effect for at least 12 months - could see tech companies fined up to A$50m ($32.5m; £25.7m) if they don't comply.
Prime Minister Anthony Albanese says the legislation is needed to protect young people from the "harms" of social media, something many parent groups have echoed.
But critics say questions over how the ban will work - and its impact on privacy and social connection - have been left unanswered.
This is not the first attempt globally to limit children's social media use, but it involves the highest age limit set by any country, and does not include exemptions for existing users or those with parental consent.
"This is a global problem and we want young Australians essentially to have a childhood," Albanese said when introducing the bill to the lower house last week. "We want parents to have peace of mind."
Having passed the Senate by 34 votes to 19 late on Thursday, the bill will return to the House of Representatives - where the government has a majority meaning it is sure to pass - for it to approve amendments, before becoming law.