Mr Trump has said he will reject any measure that does not provide funding for his proposed US-Mexico border wall.
New House Speaker Nancy Pelosi said no funding for the wall would be made available.
The Republican-run Senate said it might not even vote on the legislation.
Senate majority leader Mitch McConnell said Republicans there would not back measures that Mr Trump did not support and called the Democrats’ move a “total nonstarter” and a “political sideshow”.
The legislation passed by the House of Representatives would fund homeland security operations until 8 February and fund several other agencies until September.
How has the shutdown come about?
The partial US government shutdown began when Congress and Mr Trump failed to reach an agreement over a budget bill in December.
Mr Trump’s Republicans had passed an initial funding bill including $5bn (£4bn) for the wall, when they still had a majority in the House, but they could not get the necessary 60 votes in the 100-seat Senate.
Democrats won the majority of the House in the November mid-term elections, and the new representatives were sworn in on Thursday.
“We’re asking the president to open up government,” Ms Pelosi told the Today show before the first session of the new House. “We have given the Republicans a chance to take yes for an answer.”
The president is due to resume budget talks with top Democrats and Republicans on Friday.
First day jitters
Analysis by Anthony Zurcher, BBC Washington
The first day of a new Congress is usually like the first day of school. New arrivals wander the halls in a daze. It’s all handshakes and smiles; ceremony and sunshine.
There was a big cloud over the proceedings at the Capitol on Thursday, however. For the first time in US history, a government shutdown – the fourth-longest on record already – has stretched from one Congress into the next.
There were still receptions and flowery speeches to supporters who travelled to Washington for the occasion. But the champagne toasts were abbreviated, as the real work in Congress began almost immediately. The House Democrats scheduled votes for their legislation to reopen the government just hours after the swearing-in ceremonies wrapped up.
This was the opening move, however. Republicans in the Senate have pledged to only vote on measures that have Donald Trump’s support – and the president continues to demand his border wall funding.
As Dick Durbin, the second-ranking Democrat in the Senate, told me on Thursday, there’s “no end in sight” for the shutdown.
“I’m realistic, not optimistic,” he added.
It may be a new day and a new Congress, but hope is already a scarce commodity.
Pelosi elected as Speaker
Thursday evening’s budget vote in the House of Representatives came after Nancy Pelosi, a Democrat from California, was again elected as Speaker.
“I’m particularly proud to be a woman Speaker of the house of this Congress, which marks the 100th year of women having the right to vote,” she said.
“And, that we all have the ability and the privilege to serve with over 100 women members of Congress – the largest number in history. “
Speaking from the White House briefing room for his first time ever on Thursday, Mr Trump congratulated Ms Pelosi, saying: “It’s a very, very great achievement and hopefully we’re going to work together.”
A historically diverse Congress
As of Thursday, 102 women serve in the House, an all-time high, including 36 newly elected members and a record 43 women of colour.
While Republican women marked some firsts this past election season – like Marsha Blackburn becoming the first female Tennessee senator – the vast majority of these new lawmakers are Democrats.
Among them are the first Muslim congresswomen – Michigan’s Rashida Tlaib and Minnesota’s Ilhan Omar – and the first Native American women to serve – New Mexico’s Debra Haaland and Kansas’ Sharice Davids.
Alexandria Ocasio-Cortez of New York also became the youngest woman ever elected to Congress.
While celebrating the Democratic diversity during the swearing-in ceremonies, some on social media highlighted the contrast to the Republican members, who are mostly white men.
What does the partial shutdown mean?
- About 25% of the US federal government has no funding
- Nine departments have been affected, including Homeland Security, Justice, Housing, Agriculture, Commerce, Interior, and the Treasury
- Around 800,000 federal workers are now furloughed – that is, temporarily laid off due to a lack of funding – or working without pay
- Native American tribes who receive substantial federal funding are struggling
- National Parks have become hazardous without staff