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Swansea City and Arsenal had earlier drawn 2-2 in a dramatic match at the Liberty Stadium that featured a lightning strike from substitute Michu and a dramatic three-goal flurry in the final 10 minutes.
Sturridge, signed from European champions Chelsea, took just seven minutes to open his account on a rutted Mansfield pitch that looked more like a mudfield in places.
Suarez, who replaced the England striker in the 55th minute, made it 2-0 after just four minutes on the pitch but the goal triggered accusations of cheating after television replays suggested he had used his hand.
Neutrals hoping for an upset, in a David and Goliath clash encapsulating the magic of the world's oldest knockout competition, were given hope when Matt Green pulled one back for the Stags in the 79th and Liverpool faltered.
The game ended with Liverpool under pressure, Green having a shot turned away in stoppage time and Exodus Geohaghon rattling the visitors with his immense throw-ins.
Swansea's big Spaniard Michu made an instant impact 58 minutes into the game against Arsenal when he netted 73 seconds after coming off the bench to chants of "he scores when he wants to" from a spirited Welsh crowd.
Arsenal, without any silverware since 2005 and determined not to let their best remaining chance of the season slip through their fingers, equalised through substitute Lukas Podolski in the 81st.
Kieran Gibbs then put the visitors 2-1 up with a volley two minutes later.
The Arsenal fans' elation was short-lived, however, with Danny Graham grabbing an equaliser from close range in the 87th in a frenetic final few minutes during which either side could have produced a winner.
Arsenal's Theo Walcott had the ball in the net but was ruled offside, while team mate Laurent Koscielny fired over the bar on the rebound from a Michel Vorm save.
"A replay wasn't exactly what I was hoping for, especially with the two League Cup games against Chelsea in mind and the league games coming up," said Swansea manager Michael Laudrup.
"It's a very tight schedule. But a replay is better than a defeat."
The winners of the replay at the Emirates will face Championship (second tier) giantkillers Brighton and Hove Albion, who humbled Premier League Newcastle United on Saturday, in the fourth round.
Holders and European champions Chelsea, who thrashed Southampton 5-1 on Saturday, face lower division opponents Southend United (IV) or Brentford (III) in the next round. Liverpool now play Oldham Athletic (III).
Swansea's clash with Arsene Wenger's trophy-starved Arsenal was a slow-burner, two passing sides sparring tentatively for much of the first half before throwing caution to the wind.
Laudrup broke the deadlock by sending on Michu, who collected a header from Nathan Dyer and muscled through the defence by lifting the ball past Per Mertesacker before calmly slotting home.
It was his third goal against Arsenal this season, after scoring twice at the Emirates in the Premier League last month.
Arsenal, who won the FA Cup in 2005, were dumped out of the League Cup on penalties by fourth tier Bradford City.
With the North Londoners sixth in the Premier League, 18 points behind leaders Manchester United, the FA Cup looks their best chance of taking something from a highly erratic season.
Arsenal are also in Champions League contention, although far from favourites with a match against Bayern Munich looming next month in the last 16.