West Ham beat Leeds 2-0 in the day’s only third-round match featuring two Premier League teams while Wolves saw off Championship side Sheffield United 3-0.
Spurs needed a late flurry of goals from Harry Winks and substitutes Lucas Moura and Harry Kane to spare their blushes and beat League One club Morecambe 3-1.
Tottenham made eight changes from the side that lost to Chelsea in the first leg of their League Cup semi-final in midweek, with Kane given his first rest under Antonio Conte.
Morecambe boss Stephen Robinson was absent for the game against the club where he began his playing career after testing positive for coronavirus.
Spurs — 59 places higher than their opponents in the English league system — dominated the opening half an hour but the visitors went ahead in the 33rd minute when captain Anthony O’Connor converted from close range.
The home were level in the 74th minute but they needed a fluke — midfielder Winks overhit a free-kick from the left-hand side but it caught goalkeeper Trevor Carson out and sailed into the far corner.
Moura put Spurs ahead in the 85th minute after rounding the goalkeeper and Kane made sure Spurs would progress.
Liverpool boss Jurgen Klopp was back in the dugout at Anfield after finishing his isolation period following a positive test for Covid-19.
Liverpool named five youth team players in their team to face League One side Shrewsbury at home after a Covid-affected build-up to the match, with more academy players on the bench.
The Merseyside club have also had to deal with injuries and are without Mohamed Salah, Sadio Mane and Naby Keita, all of whom are at the Africa Cup of Nations.
Shrewsbury weathered almost half an hour of Liverpool possession but struck with their first real attack of any threat, when Daniel Udoh fired home from close range following a Nathanael Ogbeta cross.
Shrewsbury’s lead just seven minutes before Kaide Gordon scored his first goal for the club at the Kop end.
The 17-year-old controlled Conor Bradley’s low cross from the right with one foot in a crowded penalty area before slotting home with the other.
Just before half-time Liverpool were ahead when Ethan Ebanks-Landell handled in an aerial challenge with Virgil van Dijk and Fabinho scored the penalty.
Substitute Roberto Firmino increased Liverpool’s lead 12 minutes from time with a close-range backheel and Fabinho scored his second to make it 4-1 with the last kick of the game.
West Ham took the lead at the London Stadium in the 34th minute when Manuel Lanzini lashed home and Jarrod Bowen doubled the home side’s lead deep into stoppage time.
Wolves beat Sheffield United courtesy of two goals from Daniel Podence and one from Nelson Semedo while Premier League strugglers Norwich saw off third-tier Charlton 1-0.
Fourteen-times FA Cup winners Arsenal are in action at Nottingham Forest in the late-kick off on Sunday.
On Saturday, Premier League side Newcastle suffered a shock 1-0 defeat at the hands of third-tier Cambridge, while non-league sides Kidderminster and Boreham Wood also booked their places in the fourth round.