After Daniel Sturridge put Liverpool ahead with a goal on debut in the first half, Suarez extended the lead in the 59th minute by bringing the ball down with his hand before kicking it into the net.
Despite Suarez looking uncomfortable after scoring, appearing to expect the goal to be ruled out, it was awarded and proved to be decisive following Matt Green's later strike for Mansfield.
"There's no doubt it was handball but the referee and the officials ruled it wasn't deliberate," Liverpool manager Brendan Rodgers said. "I spoke to the fourth official when it was given because I couldn't see and he said it was (handball), so I can only assume they thought it wasn't deliberate."
And Rodgers said it wasn't up to Suarez to ask the referee to rule his goal out.
"It's up to the officials to decide that," Rodgers said. "That's why they get paid as officials."
The goal revived memories of the incident that overshadowed the Uruguay striker's 2010 World Cup campaign. In that instance, he deliberately handled on the goal line to block Ghana from scoring and was sent off. Ghana missed the subsequent penalty, sparking celebrations by Suarez on the sideline, and Uruguay advanced to the semifinals.
While the Ghanaians were furious with Suarez, Mansfield was more accepting.
"I'd like to give the benefit of the doubt and say it's a reaction," manager Paul Cox said, adding: "If it had gone in the other end and one of our players had done it I think we'd have accepted it ... I don't think you can call him a cheat."
But in the closing stages of the match, Mansfield fans were chanting "cheat" at Suarez, whose reputation in England took a hit in 2011 when he received an eight-match ban for racially abusing an opponent.
"He has a thick skin. He's had it throughout his career and his life in this country as a football player," Rodgers said. "That's part of his life. He deals with it remarkably well and now supporters recognize him for his qualities, which is the most important thing."
Before Suarez's handball, Sturridge looked like being the Liverpool player to grab the headlines.
Sturridge scored seven minutes into his first game since joining from Chelsea, the England striker racing onto Jonjo Shelvey's through ball before slotting it into the net.
"It was a great debut with a great goal, you can see he has that bit of pace to get in behind," Liverpool defender Jamie Carragher said.
Liverpool couldn't extend its lead against fifth-tier opposition until Suarez's contentious goal, and it was a nervy conclusion after Green powered in a late goal for Mansfield from Lee Beevers' cutback
Liverpool will play at third-tier club Oldham in the fourth round.
In Sunday's other FA Cup game, Arsenal threw away a winning position at Liberty Stadium to draw 2-2 with Premier League rival Swansea following a late goal flurry.
Former Arsenal defender Kyle Bartley had the best chance of the first half for Swansea when he headed against the crossbar, and the match only came to life after the break when the intensity increased from both teams.
It was Swansea substitute Michu who broke the deadlock after barely a minute on the pitch.
The Spaniard headed the ball past Laurent Koscielny and charged forward before slotting the ball past goalkeeper Wojciech Szczesny.
Arsenal came close to levelling when Olivier Giroud headed wide, but the goal did come in the 81st after Swansea failed to clear Theo Walcott's corner.
Bacary Sagna headed on to Laurent Koscielny, who knocked the ball through to Lukas Podolski and the Germany forward fired past goalkeeper Michel Vorm.
The visitors went in front inside two minutes after slick buildup saw Kieran Gibbs pass to Mikel Arteta, who returned the ball to the defender to blast into the net.
"Kieran Gibbs had an outstanding game," Arsenal manager Arsene Wenger said. "He gets better and better. When you see this kind of game, it's very positive."
But Arsenal held on to its lead for less than four minutes, with Danny Graham scoring after Kemy Agustien's corner wasn't headed away successfully by Podolski.
"We should have won this game ... we lacked a little bit (of) calmness and focus," Wenger said. "It's frustrating to concede a goal just like that on a corner where we couldn't clear the ball before the corner."
In the fourth round, Swansea or Arsenal will play at Brighton, the second-tier side which ousted Newcastle on Saturday.