Brendan Rodgers' side stumbled into the next round of the competition after an unconvincing win away from home over the non-league outfit, with the Uruguayan grabbing the winner
The visitors raced into an early lead with Daniel Sturridge running on to Jonjo Shelvey's superb pass before calmly curling an effort into the bottom corner, leaving Stags goalkeeper Alan Marriott with no chance.
They stretched their lead near the hour mark, with Luis Suarez tapping home after some good work by Stewart Downing down the right, with the Uruguayan forward handling the ball in the box before stroking it home into an empty net for a hugely contentious goal.
With time running out and their hopes fading, Mansfield grabbed a goal back through Matt Green after his strike inside the box took two deflections on its way in after finding himself unmarked inside the box to get something on Colin Daniel's cross.
Mansfield Town boss Paul Cox made just one change from the side that beat Alfreton Town, with winger Louis Briscoe replacing Junior Daniel in midfield.
Liverpool manager Brendan Rodgers made eight changes from the side that beat Sunderland in their last outing, with only Andre Wisdom, Lucas and Downing retaining their places while new signing Sturridge was handed his debut for the club, with Suarez dropped to the bench and captain Steven Gerrard rested completely.
After bossing possession early on, Liverpool took the lead in the 7th minute after Sturridge latched on to Shelvey's defence-splitting through ball and coolly slotting past the onrushing Marriott.
With Liverpool looking to put the tie to bed early on, Sturridge should have doubled his tally and his team's advantage shortly afterwards when the forward collected another pass from Shelvey in his stride, beating the offside trap in the process, before seeing his effort saved well by Marriott low down to his right.
The hosts had their best chance just after the half hour mark when striker Green forced Brad Jones into a smart stop with a dipping strike from outside the box after some neat work from Exodus Geohaghon down the right flank.
Sturridge went close again just before the break, bringing out a superb save from Marriott after jinking past Lee Beevers inside the box before firing a vicious strike into the bottom corner, with the goalkeeper getting a fingertip to it to push it around the post, with Liverpool going in at the break a goal ahead.
Mansfield went close straight after the interval, though, with Anthony Howell spinning inside the box after Lindon Meikle's deft pass, with Jones turning the ball away at his near post as they cranked up the pressure at the start of the second half.
Moments later, a goalmouth scramble ensued after Geohaghon's header from a corner saw Green go close with a backheel which was hacked clear off the line and Liverpool almost scored on the counter, with Sturridge screwing his effort just wide of the post in a frantic passage of play.
With the hosts chasing an equaliser, substitute Suarez grabbed the second goal in the 58th minute in controversial circumstances. After appearing to use his hand to knock the ball past Marriott after seeing his initial effort rebound up and into his path, Suarez was left to tap in.
Mansfield responded and kept their hopes of an upset alive by grabbing a goal back in the 80th minute through Green as his deflected effort flew in after Liverpool failed to clear their lines from a routine cross into the box.
This prompted an excellent response from the hosts and they began to flood forward in the pursuit of a dramatic late equaliser, with Green's snapshot from the edge of the box forcing Jones to get a strong hand to claw the ball away in the dying moments, but despite a performance full of endeavour and spirit, Liverpool held on for the narrow win.