Neither side had a single shot on target in the first half, with the match settled by a Mohamed Salah penalty in the second half. Liverpool's 50th clean sheet under Juergen Klopp carried them home.
Arsenal failed to close the gap with fourth-placed Chelsea -- who play Newcastle United in Saturday's late kick off -- as Declan Rice's first-ever West Ham goal gave his side a first home league win over Arsenal since 2006.
The defeat means Arsenal have now won just one of their last seven matches on the road in the Premier League. If Manchester United beat second-placed Tottenham Hotspur at Wembley on Sunday, they can draw level on 41 points with fifth-placed Arsenal.
There was no such slip up from Liverpool though, even if Brighton did not make it easy for the league leaders.
Roberto Firmino came close to giving his side the lead early in the first half, but could not connect with Andrew Robertson's low cross.
For all their possession, though, they were really struggling to create any meaningful chances. Pascal Gross's clumsy challenge on Salah earned Liverpool the penalty which the Egyptian converted with aplomb.
Once again, Liverpool's defence stood firm as they saw out the win -- Liverpool have conceded just 10 goals in their 22 league matches this season -- only Chelsea in 2004-05 conceded fewer after 22 matches in an English top-flight season.
"You have to work hard if you want to be successful and we were ready for it," manager Juergen Klopp said. "Be ready for the defending, be ready for the counter-press and be ready to make solutions.
"I think it was the most mature performance we have made so far. We take all the points. That's the situation. It's cool but we know that the difference tomorrow and Monday night could be different. We can only concentrate on ourselves."
ARSENAL'S STRUGGLES CONTINUE
After their 22-match unbeaten run in all competitions came to an end at Southampton in mid-December, Arsenal have struggled to find their form, with their away performances of particular concern.
On Saturday, having mustered just two shots on target against West Ham at the London Stadium, Arsenal's decline continued. West Ham comfortably held on after Rice became the first teenager to score a Premier League goal for West Ham in almost 10 years.
"It's a bad result for us," Arsenal manager Unai Emery said. "It's concerning because we need to be competitive at home and away. At home we feel good but away we are not getting the results we want."
At the bottom, Southampton continued their fine form away from home, as they earned a hard-fought 2-1 win at Leicester City, despite having played half of the match with 10 men.
Burnley's fourth win in a row lifted them to 15th in the table after they came from behind to earn a 2-1 win over fellow strugglers Fulham, while bottom-side Huddersfield Town were held to a 0-0 draw at Cardiff City.
(Reporting by Peter Hall, editing by Pritha Sarkar)
