Juergen Klopp tasted defeat for the first time as Liverpool manager after CrystalPalace's Scott Dann scored with eight minutes remaining to grab a 2-1 win at Anfield on Sunday.
Liverpool, who made a draining Europa League trip to play Rubin Kazan in Russia on Thursday, fell behind after 20 minutes when Yannick Bolasie fired powerfully home after some poor defending by the hosts.
Klopp's side struck back three minutes before halftime through Philippe Coutinho, who added to his two goals against Chelsea last weekend, when he arrived at the far post to finish a sweeping Liverpool move with a side-foot finish.
Dann, however, grabbed all three points for the visitors when he headed a corner at Liverpool keeper Simon Mignolet but reacted quickly to follow-up and finish.
Liverpool, who had won three and drawn three in Klopp's first six games in charge, are 10th in the table on 17 points, two behind Palace in eighth.
Arsenal missed the chance to top the Premier League heading into the international break when they drew 1-1 at home to Tottenham Hotspur in a high-octane London derby on Sunday.
Visitors Tottenham deservedly led at halftime thanks to a goal of stunning simplicity when Harry Kane ran on to Danny Rose's long pass forward and dispatched a shot past Petr Cech for his fifth goal in the last three league games.
Arsenal were much more lively in a frenetic second half and after Olivier Giroud had twice wasted good chances they levelled 13 minutes from time when Mesut Ozil's dangerous ball towards the far post was met by substitute Kieran Gibbs who bundled the ball past a despairing Hugo Lloris.
The draw left Arsenal in second place behind ManchesterCity who were held at bottom club Aston Villa earlier. Both clubs have 26 points with City enjoying the better goal difference.
Tottenham, unbeaten in the league since an opening-day defeat by Manchester United, moved into fifth with 21 points.
Bottom club Aston Villa finally stopped the rot by holding Premier League leaders ManchesterCity to a 0-0 draw in new manager Remi Garde's first match in charge on Sunday.
After seven consecutive league defeats, and one point from the last 30 on offer, had plunged Villa into crisis and cost Tim Sherwood his job, the former European champions produced a gritty display to frustrate City.
Villa keeper Brad Guzan made a remarkable save with his head after halftime to keep out Raheem Sterling and Kevin de Bruyne missed a sitter for the visitors who dominated possession without being able to find a way through.
The closest City came to taking the points was in stoppage time when Fernando's thumping header shook the crossbar but Villa hung on for a well-earned draw.
City have 26 points from 12 matches and will be knocked off top spot if Arsenal (25) beat Tottenham Hotspur later in the north London derby at their Emirates Stadium.
Villa remain marooned at the bottom with five points from 12 games after their worst ever start to a top-flight season.
Yet there were some positive signs for Garde, who dropped regular starters Joleon Lescott and Jack Grealish to the bench.
They played some patient football in an even first half, occasionally threatening the City goal, and defended with great determination as City later began to dominate.
"We have been playing well and we have deserved more than four points," Villa's former City defender Micah Richards told Sky Sports.
"Today is a massive draw for us.
"The new manager has come in and organised us. He has done well."
City will be kicking themselves though.
Sterling looked poised to score when he met a Jesus Navas cross from close range but his header cannoned off the face of Guzan who knew nothing about it.
Navas then slid an inviting pass across the goalmouth but De Bruyne got his legs in a tangle and failed to make proper contact with the goal gaping.