Sturridge was thrown straight into the fray by Brendan Rodgers after joining from Chelsea earlier this week and he did not disappoint, rifling home in just the seventh minute off Jonjo Shelvey's pinpoint pass.
Suarez then doubled Liverpool's lead over the Conference National side in controversial style when he appeared to handle his initial shot which was parried back at him by goalkeeper Alan Marriot before poking home.
Matt Green grabbed a late strike for the hosts to keep them in with a chance as his deflected shot found its way past Brad Jones from a deep Colin Daniel cross, but it was the Reds that progressed to the fourth round, where an away trip to Oldham awaits.
Liverpool dominated possession and saw most of the chances in the opening half at Field Mill and took only until the seventh minute to break the deadlock thanks to £12million signing Sturridge. The former Chelsea striker linked up well with Shelvey, running ahead to meet his superb ball from midfield before applying a cool finish past Mansfield keeper Alan Marriott.
That was one of several opportunities that fell to the new recruit in the opening 45 minutes, with the striker denied only by some quick-thinking saves by the bright-looking Marriott.
The Stags' back four also worked hard to keep the visitors at bay, with two last-ditch tackles denying Andre Wisdom and then Stewart Downing. Lindon Meikle was there to subdue the full-back as he charged forwards to meet Shelvey's ball, while Chris Clements did brilliantly to cut out Downing's dangerous shot.
Two bookings disrupted the run of play as the minutes trickled down to half-time, with Lee Beevers punished for tripping the lively Sturridge, while Lucas was cautioned just a minute later for holding onto Matt Green, who had forced a diving save from Brad Jones with Mansfield's best chance of the half on the 30-minute mark.
The hosts began to grow into the game in the final 10 minutes before the break, with Meikle getting a cross-come-shot away only for it to flash well wide, but it was Liverpool who had the final say as Sturridge prompted another good fingertip save from Marriott on the stroke of half-time.
Mansfield came out fighting in the second half and caught Liverpool by surprise, with Anthony Howell seeing two shots saved in quick succession soon after the restart. Louis Briscoe was also denied by Jones, while Green's close-range flick was parried by the keeper onto Joe Allen's shoulder as the Stags continued to press.
Goalscorer Sturridge was booked for a foul on John Thompson as the spell of Mansfield pressure continued and was hauled off by Rodgers soon after, with Luis Suarez replacing him, while Jordan Henderson was also introduced for Suso.
Suarez made an instant impact for the Reds with a goal that will no doubt raise eyebrows as he saw his shot blocked by Marriott then appeared to control the ball with his arm before firing home the rebound. The hosts instantly called for handball but referee Andre Marriner was unmoved.
Substitute Colin Daniel played an instrumental role in Mansfield's goal after replacing Briscoe, digging out the cross from the left that fell to Green, who rifled his effort through a crowded Liverpool box to offer the hosts a glimpse of hope. But it was not meant to be for the Stags as the Merseysiders withstood a late spell of pressure to hold on for the win.