Labour "blew the doors off" to take the Rutherglen and Hamilton West seat back from the SNP, leader Keir Starmer has said.
Speaking at a victory rally, he said it was "the first step on a very important journey" in Scotland and the UK.
The party's new MP Michael Shanks won the by-election with 17,845 votes.
This was more than double the number achieved by the SNP's Katy Loudon, with a swing of 20.4% from SNP to Labour in the Westminster poll.
The by-election was called after former SNP MP Margaret Ferrier was ousted by her constituents for breaking Covid rules.
Sir Keir told a rally in the constituency that voters had become disillusioned with both the SNP government in Scotland and the Conservative government at Westminster.
He said: "Scottish voters looked at the Tory government in Westminster and saw something that didn't represent them - they turned their backs on that.
"But they also, not so long ago, saw a Labour that had drifted away from them.
"We've changed, and because we've changed, we are now the party of change."
He added: "As for the SNP, this isn't about just a few months of turmoil in the SNP, it is about years and years of non-delivery."
Keir Starmer celebrated the election of new Labour MP Michael Shanks
Scotland's first minister Humza Yousaf said there were "a number of difficult issues around this by-election which made it a very difficult night."
The SNP leader said the "reckless actions" of their former MP during Covid and police investigations into SNP finances played a part in the poor result.
But he said his party would reflect, re-group and re-organise and bounce back stronger.
SNP candidate Katy Loudon's 8,399 votes represented a 27.6% share, down by 16.6% on the SNP result at the 2019 general election.
Michael Shanks took victory for Labour with 58.6% of the votes cast.
The newly elected MP described the result as "remarkable", and said it was the "honour of my life" to speak to thousands of voters during the campaign.
Turnout for the vote was just 37.19%, a dramatic fall from the 66.5% at the last general election.
Conservative candidate Thomas Kerr managed only a 3.9% share of the vote, a fall of more than 11% since 2019.
The chairman of the Scottish Conservatives, Craig Hoy MSP, denied that the result was "compelling evidence of a significant Labour revival".
He said: "The result was what we expected. We knew that our vote would be squeezed as the third party in a contest between Labour and the SNP."
Keir Starmer told OceanNewsUK Scotland that Labour would lay out its "positive case for change" at the party's annual conference, which begins on Sunday.
"People wanted to come out and vote for a changed Labour party," he said.
Sir Keir promised that the party would "repay that faith and trust with the change that I know they desperately want to see."
He added: "We have to earn every vote across Scotland and we will only do that if we have a positive case of change to put before Scotland and the UK.
"Having changed our party, I want to change politics."
He also used the opportunity to attack "nodding dog" Prime Minister Rishi Sunak over the cancellation of the HS2 and "circus" of the Tory conference in Manchester.
OceanNewsUK spoke to voters in the constituency after the Labour win.
Salvation Army volunteer Douglas Macdonald from Rutherglen said that he voted tactically in the by-election to get the SNP out.
The 81-year-old said: "I am most definitely happy that there was a big change in the by-election. I wanted rid of the SNP because of their reputation here in Rutherglen.
"I always vote, if you don't vote you can't complain about who gets in and what they're doing."
Douglas Macdonald voted against the SNP in Rutherglen
Conor Campbell voted for the SNP but said he was not surprised that they lost to Labour.
"It's to be expected after the recent issues with the party," he said. "I think a lot of people lost faith a little bit."
He said the SNP needed to stop pushing for a second referendum and focus on current issues in Scotland. He pointed to problems facing young people, and domestic violence and homelessness.
Kieran Paterson, 26, was one of the more than 51,000 people in the constituency who chose not to cast their vote.
"Everything that's been promised doesn't seem to manifest at all, so a vote's kind of useless at this point," he said.
The security worker said politicians need to prioritise tackling crime in the area to persuade voters like him to visit the polling station.
