David Luiz had given the visitors the lead on 24 minutes with a free-kick - his first goal for the club since returning in the summer - before a second-half Georginio Wijnaldum header brought a defiant Liverpool back into the contest.
Costa went down in the box on 76 minutes after a clip from Joel Matip, but saw the resulting spot-kick tipped behind the post by Simon Mignolet.
Both sides missed agonising chances to salvage a win in the remaining minutes, through Pedro and Roberto Firmino respectively.
Dead ball controversy
In a cagey opening quarter of an hour, Liverpool's Wijnaldum had the only clear-cut chance, forcing a save from Thibaut Courtois with a long distance strike towards the bottom corner.
But it was the league leaders who seized the initiative on 24 minutes, when David Luiz caught Liverpool napping with a swift free-kick from distance, leaving a furious Mignolet stood like a statue in the Liverpool goal.
Despite Liverpool's protests that the free-kick was taken before the referee had blown the whistle, Mark Clattenburg was having none of it.
Chelsea very nearly doubled their lead minutes later with a trademark Willian free-kick, from an almost identical position, very nearly catching the diving Mignolet off-guard.
Revitalised Reds bounce back
The second half started in Liverpool's favour, with Jurgen Klopp's side dominating every stat except the scoreline, yet it was Victor Moses who almost clinched a second for the Blues, hitting the outside of the post after being played through.
The hosts found their answer, however, just three minutes later. A Liverpool set-piece resulted in James Milner heading the ball back across goal towards Wijnaldum - via the head of Moses - who stroked home a simple header from six yards.
Costa-ly penalty miss
As both sides pursued a late winner, tensions rose and it was Joel Matip in the Liverpool box who lost his composure trying to tackle Costa, who went down easily when contact was made.
The Anfield faithful would have the last laugh, however, when the Spaniard planted a lazy penalty low to Simon Mignolet's right.
Substitute Pedro saw Chelsea's final opportunity bounce agonisingly wide of the post from long range after good individual work from N'Golo Kante.
Roberto Firmino came even closer at the other end, deep into injury time, when he headed straight at Courtois following a stunning Sadio Mane cross.
Despite frustration for both sides and an abundance of opportunities to win the game, a draw was probably a fair result in the end, with other results going in the way of league leaders Chelsea who extended their lead to nine points ahead of Saturday's visit of Arsenal.
Liverpool remain in fourth, but move within a point of both Arsenal and Tottenham in joint-second.