Senator Chuck Schumer said Democrats would support the bill if Republicans addressed a programme that shields young immigrants from deportation.
Democrats refused to vote for the bill unless they secured protections for recipients of the Obama-era programme.
Mr Schumer said the three-day shutdown was expected to end in a few hours.
"We will vote today to re-open the government to continue negotiating a global agreement," Mr Schumer said.
He lambasted President Donald Trump for failing to help reach a bipartisan deal, adding that he had not spoken with him since a meeting on Friday before the shutdown began.
"The great deal-making president sat on the sidelines," Mr Schumer said.
The New York senator said he was hopeful about talks on so-called Dreamers, more than 700,000 young immigrants brought to the US as children who were protected under the Obama-era Deferred Action for Childhood Arrivals (Daca) programme.
Republican Senate leader Mitch McConnell said his party intended to consider legislation "that would address Daca, border security and related issues, as well as disaster relief".
"We need to move forward and the first step, the very first step, is ending the shutdown," Mr McConnell said.
The Senate voted to advance the bill 81-18. Though the impasse is over, the upper chamber still has to grant final approval of the bill and send it back to the House of Representatives, where it is expected to pass.
The continuing resolution provides funding to keep the government open temporarily in the hope that Congress can reach a permanent funding agreement before the 8 February deadline.