Liverpool move two points ahead of Manchester City, who still have a game in hand, at the top.
The English champions move a point ahead of Liverpool, who host Chelsea at Anfield later on Sunday.
Having beaten Crystal Palace in December, Brighton secured their first win at Selhurst Park since 2005.
Ole Gunnar Solskjaer had to make three first-half substitutions during the 0-0 draw with Liverpool as Ander Herrera, Juan Mata and Jesse Lingard all hobbled off injured.
Palace won 2-0 away to third-tier Doncaster Rovers, the lowest league club still in the competition, while Wolves beat Championship side Bristol City 1-0.
Kieran Trippier missed a penalty before half-time as Tottenham Hotspur suffered more spot-kick woe following Thursday's shoot-out defeat by Chelsea.
Liverpool are top of the Premier League with 60 points while Manchester United are fifth with 44 points.
N'Golo Kante scored the lone goal of the match for Chelsea in the 51st minute.
Jeffrey Schlupp, Andros Townsend and Luka Milivojevic were the goal scorers for Crystal Palace.
Raheem Sterling inspired Manchester City to a 3-1 win over Bournemouth, while Crystal Palace clinched a crucial 2-0 victory over Burnley.
Tottenham secured a fifth straight Premier League away win to move just two points behind leaders Manchester City.
Alvaro Morata has now scored four in his last four Premier League games.
Unai Emery's side failed to win a 12th successive match in all competitions as they fell short of the club's record of 14 victories in a row.
Everton's third successive win in the Premier League lifted them into eighth place.
A draining schedule sees Arsenal host Leicester City at the start of three games in seven days.
Mohamed Salah was accused of diving by Crystal Palace when he tumbled in the penalty area late in the first half at Selhurst Park.
Chelsea opened their Premier League campaign with victory against Huddersfield, while Tottenham, Bournemouth, Crystal Palace and Watford also made winning starts.
After a decade in the top flight Stoke City were relegated from the Premier League after a 1-2 defeat to Crystal Palace