Joe Allen's second goal in as many matches had set a much-changed Liverpool side on its way to a home meeting with West Ham in the fourth round.
"It is a long way for these young boys but they showed what they are already able to do and what they have to learn," Liverpool manager Juergen Klopp said. "If everyone stays cool ... it is not protecting them but giving the lads time to develop."
In an all-Premier League replay, Tottenham secured its first victory over Leicester at the fourth time of asking this season.
Son Heung-min scored in the first half and then set up Nacer Chadli in the 66th minute to clinch a 2-0 victory for Tottenham, which started with top-scorer Harry Kane on the bench.
Tottenham faces third-tier Colchester in the fourth round.
"The FA Cup is an important competition for us and we showed we have a squad that is good enough to compete in every competition," Tottenham manager Mauricio Pochettino said.
Tottenham is fourth in the league, five points behind pacesetters Arsenal and Leicester.
"I am not disappointed with the performance," Leicester manager Claudio Ranieri said. "We play Tottenham four times — won one, lost one and two draws."
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