It took Liverpool debutant Daniel Sturridge just seven minutes to open his account for his new club. Jonjo Shelvey played in the former Chelsea striker, who in turn guided the ball beyond the reach of Alan Marriott.
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He had another chance with 18 minutes played, but on this occasion Marriott got the better of the England international from close range.
Next to test Marriott was Shelvey, but the Mansfield goalkeeper got both hands behind the midfielder's rising effort from 20 yards out.
Mansfield's only real chance of note arrived with 33 minutes on the clock when Matt Green saw his shot from the edge of the Liverpool penalty area parried to safety.
The hosting Blue Square Bet Premier side came out with more attacking impetus after the restart though and began to exert some real pressure on their Premier League rivals.
First Anthony Howell forced Jones into a diving save from eight yards out, then Louis Briscoe saw a shot blocked by veteran defender Jamie Carragher within the opening five minutes of the second half.
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Liverpool manager Brendan Rodgers reacted to his side coming under pressure by introducing Suarez into the action in place of Sturridge and just before the hour mark, the club's leading goalscorer doubled their advantage.
Stewart Downing squared the ball for the Uruguayan, who clearly handled the ball before beating Marriott at the second attempt. The Mansfield defence appealed for a free kick but no decision was forthcoming as Suarez ran off to celebrate with the travelling support.
However, the hosts refused to lie down and with 11 minutes left they halved the deficit through Green, who lashed a shot past Jones and Sebastian Coates on the line from the edge of the area.
Green had another chance to earn his side a shock replay in stoppage time, but Jones managed to turn his 25-yard effort away from goal.