Liverpool are still searching for their first win of the season after local rivals Everton edged the first Women's Super League match to be held at Anfield, in front of a crowd of 23,500.
Lucy Graham's long-range strike rolled into the bottom corner on the stroke of half-time after Liverpool keeper Anke Preuss had fumbled the Everton skipper's shot.
The hosts, who sit bottom of the table after registering just one point from their opening six WSL games, were the better side territorially with 11 shots to Everton's six.
Victory for Everton takes them 11 points above Liverpool and they sit in fourth place in the table.
There was huge anticipation before kick-off - a number of fans attending the 54,074-capacity stadium for the first time, many sitting in the famous Kop stand.
The home fans sang 'You'll Never Walk Alone' as the players walked out on to the pitch but you could hear the Everton contingent chanting throughout.
Those 'noisy neighbours' were almost silenced within two minutes when Melissa Lawley's long-range shot was saved by Everton keeper Tinja-Riikka Korpela and Liverpool came close again moments later when Niamh Charles' cut-back almost fell for Courtney Sweetman-Kirk.
Twenty-year-old Charles, a lifelong Liverpool fan who regularly attends matches at Anfield to watch Jürgen Klopp's Reds, was the orchestrator of a number of chances in the first half.
Kirsty Linnett was the most frequent beneficiary and thought she had broken the deadlock, only to be denied by a sublime save at the right-hand post by keeper Korpela.
Liverpool's dominance ultimately came to nothing and they were punished for it when Preuss' costly mistake gifted Everton a goal seconds before the half-time whistle.
The second half continued in much the same manner as Charles' shot was blocked in the six-yard area, Danielle Turner sliding in at the crucial moment.
Everton, though, always posed a threat on the break and Simone Magill could have doubled their lead late on but her first-time volley bounced inches wide.
The visiting fans behind the goal roared in celebration moments later when the full-time whistle went and the players huddled together before thanking the supporters.
While Liverpool's winless run extends to six games in the league, Everton boss Willie Kirk has got his team firing with four wins, including two on the road.
But the Reds, who had an average attendance of 536 last season at their usual ground - Tranmere Rovers' Prenton Park - will be pleased with the turnout on a weekend where women's football has been celebrated at several men's grounds across the UK.
Liverpool manager Vicky Jepson told BBC Radio 5 Live:
"I'm devastated for the type of goal we conceded, because they weren't going to score unless we give them that goal.
"I know Anke is going to be gutted about that and we'll all look after her and pick her up from the mistake.
"In terms of our own team performance, it probably hurts more than ever as we were the better team. I'm not just saying that because of the occasion.
"We created so many opportunities, but we just couldn't find the back of the net."
Everton manager Willie Kirk told BBC Radio 5 Live:
"As a club, we've not had much success here over the years. Whether it's men, women or under-10s, any Merseyside derby win is special.
"To do it here in front of 23,500 people, it's a special day. The players are buzzing as are the staff for the way they prepared the team.
"I've not tried to dampen the occasion this week, I've refused to talk it down. I wanted the players to feel the buzz and get inspired by it.
"I talked about winning a 90-minute performance and we've got the three points and it's put us in a great position."
On climbing to fourth in the table: "We're not title contenders, we finished second from bottom last year.
"We've brought a lot of new faces in over the summer and it's brought some great competition into the club. But we've probably got the 10th-highest budget in the league.
"If we can finish in the top half this season, that for us will be seen as progress, so we won't get carried away."