The hosts - 93 places below their Premier League opponents in the league system - fell behind early on when Daniel Sturridge netted on his Liverpool debut.
Luis Suarez doubled their advantage in controversial circumstances, appearing to control the ball with his hand before smashing home from close range.
Mansfield rallied and set up a grandstand finish when Matthew Green scored via the aid of two deflections, but Liverpool held on to secure a fourth-round tie against Oldham Athletic later this month.
Here, Sports Mole takes a look at the Reds' display and whether the hosts deserved more on a day full of incident in a classic cup tie.
On Target 13
On Target 10
Was the result fair?
Liverpool enjoyed the majority of possession but Mansfield's magnificent response in the second half can leave little argument that a replay would have been merited. Quality in the final third eventually told for the visitors, who will be disappointed to have conceded so many free kicks.
The non-league side were slow to start but reacted well to going behind and came out fighting in the second half. Paul Cox's men even kept going following the blow of conceding a dubious second goal and can be more than proud of their efforts this afternoon.
A strong first-half display had the Reds cruising but Rodgers's side took their foot off the gas after the interval. Jonjo Shelvey was impressive for long spells, while Suarez and Sturridge both showed glimpses of their undoubted quality.
Sports Mole's man of the match
Shelvey The young midfielder dropped off into dangerous areas throughout and provided a consistent supply of chances for Sturridge in the opening half. Intelligent on the ball and willing to work hard off it, he can be pleased with his performance on a difficult pitch.
Andre Marriner will take some flack for allowing Suarez's contentious second to stand, but he was quick to give decisions and did not allow the flow of the game to be disrupted at any point.
Mansfield: It's back down to earth for the home side as they entertain Kidderminster on Saturday.
Liverpool: Liverpool come up against an in-form Manchester United side at Old Trafford next weekend in a Premier League clash with an edge.