However, the draw did extend Chelsea's lead at top of the Premier League to nine points.
Liverpool keeper Mignolet made up for his first-half embarrassment when he was beaten by David Luiz's superb free-kick from 25 yards.
Georginio Wijnaldum's close-range header in the second-half gave Liverpool a draw they fully deserved, ending a run of three home losses in a row.
However, it could have been much better for Chelsea and worse for Liverpool when Costa went to ground 14 minutes from the end.
Referee Mark Clattenburg pointed to the spot but Mignolet dived low to his right to save the Spain striker's spot-kick.
Arsenal 1-2 Watford: Arsenal's Premier League title hopes suffered a huge blow with a shock home defeat as Watford secured their first top-flight win over the Gunners since 1988.
Bournemouth 0-2 Crystal Palace: Crystal Palace finally earned a Premier League victory under Sam Allardyce as goals by Scott Dann and Christian Benteke secured the points against Bournemouth away at a rain-swept Vitality Stadium.
Burnley 1-0 Leicester City: Burnley secured their fifth home Premier League win in a row with victory over Leicester at Turf Moor, thanks to Sam Vokes' late goal.
Middlesbrough 1-1 West Bromwich Albion: Middlesbrough came back from a goal down but were left frustrated as a hard-working West Brom claimed a point at the Riverside Stadium.
Swansea City 2-1 Southampton: Swansea boosted their Premier League survival hopes and secured back-to-back wins for the first time this season with a spirited victory over Southampton.
Sunderland 0-0 Tottenham: Title-chasing Tottenham had to settle for a point at the Stadium of Light as Sunderland climbed off the bottom of the table with a hard-fought draw.
Chelsea had to settle for a draw against a strong Liverpool side at Anfield after Diego Costa's late penalty was saved by Simon Mignolet.