It was a late late show at Anfield as the reds, 1-0 down, scored in the 86th and 95th minute to seal an important victory.
Jordan Henderson was named as captain with several first team players rested. Brendan Rodgers made 9 changes from the team that drew against Hull on Saturday. Lambert played up front, and Borini and Markovic were handed starts on the wings, Coutinho occupying the centre.
Liverpool's passing was sharper and crisper than in recent weeks as they dominated the majority of the first half. Markovic guilty of the miss of the half when he sliced a shot well wide from inside the box.
Coutinho and Borini were involved in everything good about Liverpool, although in the final ten minutes of the half the reds did cede plenty of possession, corners and free kicks in dangerous areas.
The second half and the reds started slowly - slowly enough that Swansea gained the advantage. In the 65th minute Emnes found himself with time and space at the edge of the 6 yard box and he volleyed past Brad Jones.
Liverpool huffed and puffed. Lallana replaced Markovic, Balotelli replaced Lambert. Borini was still grafting and with 4 minutes of normal time remaining his exquisite cross was met by the foot of a jumping Mario Balotelli. The Italian Equaliser.
Federico Fernandez would be handed a harsh red card for a foul on Coutinho as the 5 minutes of injury time got underway.
As the minutes ticked down, Coutinho sent in a free kick - Dejan Lovren, partly at fault for Swansea's goal, rose at the back post and headed down and in to send Anfield into jubilation.
Liverpool are now one of 4, possibly 5, Premier League sides in the Quarter Finals - thanks in large part to Balotelli, Lovren and Fabio Borini, the man that the reds wanted to get rid of.
Next up for Liverpool - a weekend trip to Newcastle.
LFC Reds Man of the Match: Fabio Borini (44.8%) Match Topic
