The package would mark the biggest tax overhaul since the 1980s. It was passed by 51 votes to 49, after a series of amendments in a marathon session.
Democrats complained it only benefited the wealthy and big business.
The plan sees a sharp cut in corporation tax, but a Senate committee finding has warned it would add $1tn (£742bn) to the budget deficit.
President Trump wants the measures enacted by the end of the year and congratulated Republicans for taking the US “one step closer to delivering massive tax cuts for working families”.
The Senate will now have to merge its legislation with that passed last month by the House of Representatives, before it can be signed into law by the president.
The move will be seen as a major victory for Mr Trump, who since taking office has struggled to get major legislative movement in Congress – including fulfilling his vow to repeal and replace Obamacare.
His presidency has also been dogged by an independent investigation into Russian attempts to influence the 2016 US election and possible collusion with his campaign team. On Friday, ex-national security adviser Michael Flynn became the Trump administration’s most senior member to be charged in the investigation.
Republican legislative success at last
Analysis: Anthony Zurcher, BBC News, Washington
The US Senate, a seemingly insurmountable roadblock for the Republican agenda for much of this year, has at last given its assent to a major piece of legislation.
Perhaps unsurprisingly it was sweeping tax cuts – always beloved by conservatives – that finally brought the party together and gave President Donald Trump the opportunity to claim a landmark legislative achievement.
It wasn’t always pretty. Senate negotiators were handwriting amendments to the massive bill practically up until the final votes were taken. Deals within deals were cut to satisfy recalcitrant legislators. Democrats howled at the permanent cuts provided to corporations, while middle-class taxpayer benefits had sunset clauses.
In the end, however, a combination of hope and fear were enough to drag a slim majority into the yes column. The hope is that a booming economy will give Republicans a chance to run on their tax policy when they stand before voters next November. The fear was that yet another failure would have led to a revolt among the party’s big donors and traditional business constituencies.
The House and Senate must now smooth out differences in their bills and vote on the compromise. It’s not the end of the race, but the finish line is in sight.
Presiding over the Senate, Vice-President Mike Pence declared the 51-49 victory to applause from Republicans in the early hours of Saturday morning.
The new tax plan would see the corporate tax rate lowered from 35% to 20%, and include more modest tax cuts for individuals across income levels.
Following the vote, Senate Democratic leader Chuck Schumer said his opponents would pay the price at the ballot box in mid-term elections next year.
“My Republican friends will ultimately pay consequences for this bill in 2018 and beyond. The Republican party will never again be the party of tax cuts for middle-class people,” he warned.
He said the measures would endanger social security and medical provision.
Republican Senate majority leader Mitch McConnell said the legislation would prove to be “just what the country needs to get growing again”.
He brushed aside complaints that it was pushed through without proper scrutiny, saying: “Everybody had plenty of opportunity to see the measure. You complain about process when you’re losing and that’s what you heard on the floor tonight.”
Republican Senator Bill Cassidy also praised the bill, saying “working families and middle-income families across the nation will be better off”.
The final draft of the Tax Cuts and Jobs Act bill went through several changes on Friday in order to bring reluctant Republicans on board. Republicans have a 52-48 majority in the Senate.
Democrats were angry about the last-minute revisions, complaining that they had not been given enough time to digest the nearly 500-page document, with handwritten changes to the legislation.
The only Republican senator who refused to back the legislation was Bob Corker.
“I am not able to cast aside my fiscal concerns and vote for legislation that… could deepen the debt burden on future generations,” he said.
On Thursday, the non-partisan Senate Joint Committee on Taxation found that the Tax Cuts and Jobs Act would add significantly to the federal deficit over a decade.
The report contradicted a White House claim that economic growth would compensate for the tax cuts.
BBC contributed to this report.