Brendan Rodgers's side, who had failed to win their last five games in all competitions, eventually secured a 3-2 shootout win at Anfield when Bastien Hery's tame effort for Carlisle was easily saved by Adam Bogdan.
The Merseysiders had taken an early lead in the game through Danny Ings but were pegged back by a goal from Derek Asamoah after 34 minutes then could not find a breakthrough despite battering at the door and missing a series of chances.
Arsenal's Mathieu Flamini proved an unlikely hero with two goals as they knocked bitter north London rivals Tottenham Hotspur out with a 2-1 victory.
Holding midfielder Flamini poached the first from close range in the first half and then hit a superb volley on the run after the restart after an own goal from Calum Chambers had levelled the tie at White Hart Lane.
Holders Chelsea eased into the fourth round with a 4-1 win over third tier Walsall, while Southampton thrashed second tier Milton Keynes Dons 6-0 with two goals each for Jay Rodriguez, Sadio Mane and Shane Long.
Newcastle United's dismal start to the season continued as they were beaten 1-0 at home by second division Sheffield Wednesday, while Manchester United made light work of Ipswich Town, winning 3-0 at Old Trafford.
Crystal Palace won 4-1 at home to Charlton and Norwich City won an all-Premier League clash against West Bromwich Albion 3-0.
