Liverpool new-boy Daniel Sturridge made an immediate impact as his side saw off a spirited Mansfield effort to win 2-1.
Sturridge, a £12million buy from Chelsea a few days ago, wasted no time in getting off the mark and helping the Reds into the fourth round - and another tricky trip to Oldham - at the expense of their non-league opponents.
The Blue Square Bet Conference side had their moments, notably at the start of the second half, before Luis Suarez further crushed their hopes in the 59th minute - and not without controversy. If a contentious handball had been spotted then perhaps Matt Green's 79th-minute strike might have forced a lucrative replay at Anfield, but it was not to be.
It took just eight minutes for Liverpool, and Sturridge in particular, to make their mark. The Stags defence was scythed open with ease by a delightful through ball from Jonjo Shelvey, and with Sturridge showing a turn of speed in racing on to the pass, he slotted comfortably beyond Alan Marriott from 12 yards.
Within 90 seconds it should have been 2-0, and effectively game over, as the Shelvey-Sturridge partnership combined again, with the latter clean through on to another incisive defence-splitting pass. On this occasion, however, Sturridge dithered perhaps a fraction too long and shimmied once too often as his eventual crack at goal was beaten away by Marriott at close range.
With Suarez on the bench, Sturridge was given his chance to shine and a minute before the break he again managed to get in behind the home defence from a Shelvey pass, firing in a shot Marriott touched beyond the far post.
Whatever Mansfield boss Paul Cox said to his players during the interval worked a treat as Town gave the Reds the fright of their lives in the opening few minutes of the second half as they besieged their opponent's area. It was all sparked by a close-range shot from Anthony Howell that Brad Jones nudged past his right-hand post.
It was hair-raising stuff for Liverpool, who eventually drew breath in the 53rd minute on a breakaway which culminated in Sturridge dragging a Downing pass across Marriott. It proved to be his last act of his debut as Rodgers replaced the 23-year-old and Suso moments later with Suarez and Jordan Henderson.
The Uruguayan's introduction paid immediate dividends, but not without typical debate for which he has become renowned. Again Downing was the provider and after Suarez's initial effort was blocked by Marriott, he then used his hand to bundle home the rebound from a couple of yards, sparking appeals from the Town players that went unrewarded.
The game appeared to be petering out towards a natural conclusion until Green struck 11 minutes from time with a right-foot shot from 15 yards that bounced off the turf and beyond Jones, heralding a frenetic finish. The most notable chance saw Matt Green lash a shot goalwards from 20 yards in the first of five minutes of injury time, only for Jones to paw away.