Lib Dem leader Tim Farron claims his party is "back in the big time" after it fought on the issue of Brexit to oust ex-Conservative MP Zac Goldsmith in the Richmond Park by-election.
Lib Dem challenger Sarah Olney overturned Mr Goldsmith's 23,015 majority to finish 1,872 votes ahead.
Mr Goldsmith quit the Tories to stand as an independent after the government backed a third Heathrow runway.
But the Lib Dems successfully switched the focus of the campaign to Brexit.
Mr Goldsmith was among Conservatives who backed a Leave vote in the 23 June referendum on the UK's membership of the EU.
Mr Farron said the outcome was a verdict on Theresa May's "UKIP-ish" take on Brexit and showed that it was possible for moderates to win.
The result was seized on by the European Parliament's lead Brexit negotiator Guy Verhofstadt - a Liberal politician - who congratulated Ms Olney and said Europe "is watching".
But Mr Verhofstadt's intervention provoked anger from Conservative former leader and prominent Leave campaigner Iain Duncan Smith who warned him to "mind his own bloody business".
He told The Telegraph: "I assume this means he is prepared to stand by the result of the EU referendum which means that we are leaving, so he should stop moaning and carping and trying to damage this."
Conservative Remain campaigner Anna Soubry said the result was "sensational", tweeting that MPs "ignore Remainers at their peril & u can forget #Hardbrexit".
Mr Farron told BBC2's Daily Politics: "Overturning a 23,000 majority comes as a shock under any circumstances."
He added: "Sarah is a reminder that populism does not automatically have to win. Those results: Brexit, Trump, the last general election, there was nothing inevitable about them.
"It is entirely possible for moderate progressive people to win.
"This is the beginning, not just of the British people choosing to own what happens next over Brexit - it's also the beginning of Britain finally having a decent, moderate, tolerant opposition to the Tories that fills the space that Jeremy Corbyn's Labour party have left behind.
"This is the beginning of something special... This is a historic moment for the country - let's see what happens next."
But John Curtice, professor of politics at the University of Strathclyde and president of the British Polling Council, urged caution at reading too much into the result.
While it did give the Lib Dems a chance to demonstrate their credibility, and get themselves noticed in what had previously been a Lib Dem seat, he told BBC Radio 4's World at One programme: "To try to suggest that this by-election success tells us anything about the mood of the country in relation to Brexit is mistaken.
"The polling evidence is that we still have a situation where the country is divided pretty much 50:50 on the merits of Remain versus Leave, much as it was on 24 June."
Lib Dems poured resources into the contest in the staunchly Remain south-west London seat after Mr Goldsmith quit as a Conservative MP to force the by-election, so he could run as an anti-Heathrow expansion independent.
Who is Sarah Olney?
A relative newcomer to politics, Sarah Olney said she was compelled to join the Liberal Democrats after the 2015 general election.
In her victory speech, she said: "A year and a half ago I wasn't involved in politics, I wasn't a member of a political party, I had never been involved in a political campaign, I had never thought about being a politician.
"But I knew I was a liberal.
"When I saw what happened in the general election last year I felt I had to get involved."
Ms Olney, who lives in Kingston with her husband Ben and two children, works as a qualified accountant at the National Physical Laboratory in Teddington.
The vote saw Labour's Christian Wolmar losing his £500 deposit as he trailed a distant third. The other five candidates also lost their deposits as they did not receive a big enough share of the vote.
Ms Olney, who also opposed Heathrow expansion, will be the Lib Dems' only female MP - joining eight male colleagues.
She said the by-election result was a rejection of the "politics of anger and division" and pledged to vote against Article 50 - the legal process that sparks Britain's exit from the EU - if it is put to a parliamentary vote.
"That's been a central part of my campaign and now I've been given a very clear mandate that that's what they [constituents] want me to do," she told the BBC.
But Ms Olney disappeared off air when she was grilled further over her stance on Brexit during an interview on Talk Radio. An unnamed aide stepped in and told presenter Julia Hartley-Brewer: "I'm really sorry but Sarah has to leave now."
Earlier, Housing Minister Gavin Barwell said the by-election result "doesn't change anything in terms of government".
And Downing Street added: "The referendum result was very clear and the majority of the country expressed an opinion for us to leave the EU.
"The government is getting on with delivering that."
Conservative MP Jacob Rees-Mogg said he believed the vote had been about Brexit as Richmond "was a very strong pro-Remain constituency".
He praised Mr Goldsmith's "nobility" for keeping his word, but said it was a "real shame" that an independently minded person like him would no longer be in Parliament.
After his defeat was announced, Mr Goldsmith said: "This by-election was not a political calculation.
"It was a promise that I made and it was a promise that I kept.
"I wish Sarah well in her very, very important job and I hope she serves this community as well as this community deserves."
- Sarah Olney (Liberal Democrats) - 20,510
- Zac Goldsmith (Independent) - 18,638
- Christian Wolmar (Labour Party) - 1,515
- Howling Laud Hope (The Official Monster Raving Loony Party) - 184
- Fiona Natasha Syms, (Independent) - 173
- Dominic Francis Stockford, (Christian Peoples Alliance) - 164
- Maharaja Jammu and Kashmir (One Love Party) - 67
- David Powell - 32
Turnout = 53.6%
Ms Olney was elected with a majority of 1,872 votes, compared with a Conservative Party majority of 23,015 at the 2015 general election.