Goals from Luis Suarez and Jordan Henderson had given the Merseysiders a healthy advantage heading into the final 30 minutes of the game, but Olivier Giroud and Theo Walcott both beat Pepe Reina with emphatic finishes to set up a pulsating final 20 minutes.
Both sides had chances to score a late winner - Giroud being denied by Reina in his side's closest attempt, while Szczesny was lucky to keep out a very late Suarez attempt - but neither could convert, as the game finished level.
A late strike from Gareth Bale grabbed Tottenham a share of the points in a 1-1 draw against Norwich at Carrow Road.
Andre Villas-Boas's side fell behind after 32 minutes to Wesley Hoolahan's strike, but with ten minutes remaining Bale rescued a point - getting the ball from Jermain Defoe before running with it and unleashing a hammering left-footed shot that was beyond Mark Bunn before he could get down to it.
Elsewhere, a brace from Leighton Baines was enough for Everton to stave off a spirited West Brom fightback and claim a 2-1 win at Goodison Park - moving them within a point of fourth-placed Spurs.
Baines opened the scoring on the half-hour, and in fine fashion too. After receiving the ball from Leon Osman, Baines used his trusty left-foot to rifle the ball beyond Ben Foster through a crowd of players.
In injury time at the end of the first-half he made it 2-0, this time from the penalty spot. Claudio Yacob was the guilty party for the Baggies - bringing down Victor Anichebe - and Foster was unable to bail his midfielder out, as Baines duly converted.
The Baggies created few clear-cut chances in the game but they were given a lifeline after 65 minutes, as Shane Long headed home from close range. An equaliser would not be forthcoming, however, as the Toffees improved their bid for Champions League football.