Luiz gave Chelsea the lead in the first half but Liverpool managed to draw level thanks to Wijnaldum's 57th minute header
Arsenal suffered a shock 2-1 home loss to Watford who managed a first league win in eight games
Champions Leicester City's relegation worries are becoming acute after Claudio Ranieri's side went down 1-0 at Burnley
Diego Costa's missed second-half penalty cost Premier League leaders Chelsea a victory at Liverpool on Tuesday but a 1-1 draw did little damage to their title prospects as closest rivals Arsenal and Tottenham Hotspur dropped points.
Spain striker Costa was denied by Liverpool keeper Simon Mignolet, who made amends for being caught off guard by a superb David Luiz free kick midway through the first half.
Georginio Wijnaldum's close-range header levelled the scores in the 57th minute as Liverpool avoided a fourth consecutive home defeat in all competitions, although Juergen Klopp's side remain 10 points adrift in fourth place.
"It's important to draw against a really good team and now we must focus on playing Arsenal next week," Chelsea manager Antonio Conte said ahead of Saturday's game at Stamford Bridge.
Arsenal, who began the day second, must recover quickly from a shock 2-1 home loss to Watford who managed a first league win in eight games and celebrated their first top-flight victory over the Gunners since 1988.
Watford scored twice inside the first 13 minutes through Younes Kaboul and Troy Deeney and hung on for the victory after Alex Iwobi pulled one back near the hour mark.
"It was obvious we lost duels and were not sharp enough. It looked more mentally that we were not ready for the challenges," said Arsenal manager Arsene Wenger, who was serving a touchline ban and had to watch gloomily from the stands.
Arsenal are nine points behind Chelsea, who have 56, along with Tottenham Hotspur who moved above their north London rivals on goal difference after a lacklustre 0-0 draw at basement club Sunderland. Spurs also lost defender Danny Rose to injury.
Champions Leicester City's relegation worries are becoming acute after Claudio Ranieri's side went down 1-0 at Burnley for whom Sam Vokes grabbed a late winner.
Leicester's third consecutive league defeat left them two points off the relegation zone in 16th place with the battle for survival hotting up after wins for Swansea City and Crystal Palace and Sunderland's battling point at home to Spurs.
Third-bottom Palace won in the league for the first time under new manager Sam Allardyce as Scott Dann and Christian Benteke scored the goals in a 2-0 victory at Bournemouth.
Gylfi Sigurdsson's 70th minute effort earned Swansea a second consecutive win to keep them just above the trapdoor.
Sliding Middlesbrough eeked out a point in a 1-1 draw at home to West Bromwich Albion but remain in trouble.
On Wednesday, fifth-placed Manchester City, 13 points off the pace, visit West Ham United while Manchester United, a further two points back, host Hull City, who are now bottom.
Chelsea have not lost a league game at Anfield since 2012, although they will probably feel a little short-changed after failing to capitalise on Liverpool's lack of form.
Klopp's side will be glad to see the back of January having won just once in nine games -- an FA Cup replay against Plymouth Argyle -- and suffering three consecutive home defeats.
After Liverpool made a bright start at Anfield it looked like another night of woe when the crafty Luiz noticed Mignolet arranging his wall and seized his chance to plant a superb free kick into the corner left vacated by the distracted keeper.
There was plenty of spirit from Liverpool, though, and Wijnaldum levelled in the 57th minute after James Milner did well to head Jordan Henderson's cross back towards goal.
When Costa went down after a challenge by Joel Matip 14 minutes from time it looked as though Chelsea would avenge their defeat at Stamford Bridge earlier this season but Mignolet spared the hosts with a fine
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