The measure passed 51-50 on a party line vote with Vice President Kamala Harris casting her first-ever tie-breaking vote to tip the balance in the early hours of Friday morning.
The approval allows Democrats to push ahead with the budget reconciliation process, enabling the party to pass the relief package without Republican support.
The passage came after hours of votes on dozens of amendments, and the House will now have to vote again on the separate version after passing the initial version on Wednesday.
After meeting with Mr Biden at the White House on Friday morning, Speaker Nancy Pelosi vowed to pass the bill in the next two weeks, and pledged further action beyond that to help the American people.
"This is the American Rescue Plan," she said. "The next bill will be the recovery plan."
While many of the GOP amendments were rejected, one significant amendment came from a bipartisan group of senators led by Democratic Sen Joe Manchin and Republican Sen Susan Collins.
The group had been pushing back against certain provisions for stimulus checks in the bill, claiming the plan provides too much money to high-income Americans.
An amendment proposed by the group that would prevent "upper-income taxpayers" from being eligible to receive $1,400 Covid relief checks passed 99-1.
Notably, the amendment is not binding and it does not mean that the eligibility requirements will be changed in the final package.
Other amendments included rescue grants for restaurants and one to prevent tax increases on small businesses amid the ongoing health crisis.
"We have moved forward. Many bipartisan amendments were adopted. ...This was a giant first step," Senate Majority Leader Charles Schumer said following the vote.
"We will keep working as hard as we can to pass this legislation through the House, through the Senate as we go through the reconciliation process and hopefully put it on the president's desk."
He would like to win “support from Republicans” for his $1.9trn aid bill “if we can get it,” he said when asked about using the budget reconciliation process to bypass Republican obstruction.
But he added: “The Covid relief has to pass. … There’s no ifs, ands or buts.”
Ten Republican senators who met with Mr Biden on Monday sent a letter to the White House on Thursday raising concerns over the size of the proposal, particularly in relation to funding for K-12 schools.
Democrats will need near-universal support from House Democrats and a unanimous vote of the 50-Democratic members of the Senate to pass the bill without Republican support.
Ms Pelosi told reporters at the White House on Friday morning that a House reconciliation bill will be introduced on Monday and she hopes for bipartisan support.
The speaker says the goal after congressional committee reviews and House procedures is to send the revised version of the relief package to the Senate within a two week timeframe.
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