Everton’s encouraging start to the Women’s Super League season continued with a victory over their neighbours in a derby at Anfield – something their male counterparts have been unable to manage so far this millennium.

A single goal at the end of the first half from the captain Lucy Graham allowed the visitors to climb to fourth in the table and left a Liverpool side still looking for their first win of the season stuck at the bottom with a single point. That was actually harsh on a home side who produced many of the game’s best moments only to run up against an Everton defence in miserly mood.

“When you have a defence like ours, you rarely concede from open play,” Everton’s goalscorer said. “Liverpool played well, I must admit, but I’m really happy for our fans because they’ve finally seen an Everton win at Anfield.” The crucial goal was something that will haunt the Liverpool goalkeeper Anke Preuss for a while, but Willie Kirk said he had seen his captain strike from a similar distance before. “She scored a goal like that against us last season,” said the Everton manager . “That’s one of the reasons we brought her in.”

Liverpool produced most of the early attacking and it took an agile save from Tinja Korpela to keep out a goalbound header from Kirsty Linnett a quarter of an hour into the game. Mel Lawley and Niamh Charles looked lively for the home side in the first half-hour, though Liverpool could not quite turn their possessional superiority into clearcut chances, with Kika van Es and Gabby George solid at the heart of the Everton defence.

Linnett and Charles sent hopeful efforts narrowly wide as the interval approached, with Everton content to stay compact at the back and leave Elise Hughes foraging alone up front, though a long-range effort from Linnett brought a save from Korpela on the stroke of half-time. When a promising passing move at the end of the first half saw Courtney Sweetman-Kirk, Linnett and Mel Lawley move the ball along the edge of the Everton area to allow Charles the briefest of shooting chances, she turned into trouble instead of going for goal and the opportunity was lost.

Quick Guide Women's Super League roundup: Chelsea stay top Show Georgia Stanway enjoyed a busy afternoon, scoring two goals before being sent off for a second booking during Manchester City’s 5-0 evisceration of West Ham, which took them to the top of the table. But only briefly: Chelsea’s nervy win over Manchester United – Maren Mjelde scoring the game’s only goal from the spot – restored the Blues’ one-point lead later in the day. Ebony Salmon’s 83rd-minute equaliser capped a thriller at the Madjeski, where Reading and Bristol City shared six goals, while on the south coast Kayleigh Green scored either side of a Léa Le Garrec wondergoal to ease Brighton towards a 3-0 win over Birmingham City. Alex Hess Photograph: Harriet Lander - Chelsea FC/Chelsea FC

Just when Liverpool thought the first half had been safely negotiated, a speculative attempt from Graham from well outside the area put Everton unexpectedly in front. Preuss in the Liverpool goal had been given almost nothing to do for 45 minutes, but although she had Graham’s effort covered she let the ball slip through her hands and into the net.

Both the half-time team talks were turned on their head, for suddenly Liverpool had nothing to show for their superiority while Everton held a lead they barely deserved and had hardly even been looking for. “I was aware of the time on the clock and I just wanted to get in a shot on target.” Graham said. “To be honest we weren’t performing all that well before half-time.”

When a rare mistake by Abbey-Leigh Stringer allowed Sweetman-Kirk a shooting chance at the start of the second half the Liverpool striker’s effort cleared the bar. Everton went straight down the pitch to set up an attack of their own and it took a well-timed cover tackle from Leighanne Robe to prevent Hannah Cain getting in a shot on goal.

Chantelle Boye-Hlorkah tested Preuss with a shot after an hour, a better struck effort than the one that had brought the Everton goal, but found the Liverpool goalkeeper right behind the ball. Charles had the next chance in what was now an end-to-end contest, lining up a shot at goal after substitute Rinsole Babajide had made great progress down the left, only to find Danielle Turner blocking the effort with an exceptional cover tackle.

Babajide made a difference when she came on, making inroads down both flanks and bringing a save from Korpela, though with Charles shooting narrowly wide in the closing minutes Liverpool were not quite able to find the equaliser they sought. Exceptionally tricky with the ball at her feet, Babajide nevertheless wasted the last chance of the afternoon, shooting too high from an optimistic distance with teammates up in support. Everton could have scored a second goal right at the death, Molly Pike putting a shot wide from Cain’s cross as Liverpool were caught with too many players in upfield positions, though a two-goal margin of victory really would have flattered the visitors.

“Defeat was tough to take because we were the better team today,” said the Liverpool manager, Vicky Jepson. “I don’t think they would have scored but for us gifting them a goal. We’ll come back from this, but at least a crowd of 23,000 saw us playing well. That’s the cause for celebration because I’d like to think a lot of those people will be tempted to come and watch us again.”