Crystal Palace 1-2 Liverpool Roberto Firmino scored a late winner as Liverpool maintained their eight-point lead at the top of the Premier League with a hard-fought victory over Crystal Palace. The Brazil striker poked in from a corner five minutes from time to restore his side's lead three minutes after it looked as if Wilfried Zaha had rescued a point for the Eagles. Zaha's 82nd-minute goal was his first of the season and cancelled out Sadio Mane's 10th goal of the season just after half-time. Palace winger Zaha then wasted a great chance to score a second equaliser late on when he blazed over from close range. Palace defender James Tomkins also had a goal ruled out in first-half stoppage time by the video assistant referee after Jordan Ayew was adjudged to have fouled Dejan Lovren. Liverpool's victory means they have now gone 30 games unbeaten in the Premier League. Leicester City 2-0 Brighton Jamie Vardy maintained his brilliant scoring streak as Leicester beat Brighton for their fifth successive Premier League win to strengthen their hold on second spot. The Foxes remain eight points behind leaders Liverpool, but are now three clear of third-placed Chelsea and four points above Manchester City before those teams meet later on Saturday. Vardy has been key to the Foxes' form and he once again was the shining light on a dark, rainy day on the south coast. The 32-year-old set up Ayoze Perez for the 64th-minute opener after swift counter-attack and then added the second from the spot, after his initial spot-kick, which was saved, had to be retaken for encroachment. Foxes goalkeeper Kasper Schmeichel failed to be tested once by the home side, although Wilfred Ndidi made a brilliant block to deny Martin Montoya from hitting the target. Arsenal 2-2 Southampton Arsenal drew against Southampton to increase the pressure on manager Unai Emery, who saw his side slip to ninth in the Premier League. The result leaves the Gunners nine points worse off than at the corresponding stage last term, while the Saints remain 19th. Emery's side, who were second best throughout, were booed off at the final whistle after delivering a disjointed performance which suggested they may struggle to compete for a place in the top four this season. Danny Ings' early goal had put the visitors in front, with the forward catching the home defence asleep to race onto Ryan Bertrand's free-kick and score with a right-foot shot from close range. That set the tone for what was to follow, with the visitors wasting several good chances to extend their lead before Alexandre Lacazette turned in a close-range equaliser. The goal did little to kick-start Arsenal though, who fell behind for a second time when James Ward-Prowse scored on the rebound after seeing his second-half penalty saved by home goalkeeper Bernd Leno. Not even Lacazette's second goal of the day, which came late in stoppage time, could raise spirits at the Emirates, with the forward appearing embarrassed to celebrate.