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.
