Playing at home for only the second time after their Brunton Park home was flooded, Carlisle were hoping to add to Everton's woes after the Merseysiders' League Cup semi-final defeat by Manchester City in the week.
But after Kone tapped in Lennon's cross in the second minute, the visitors did not look back, with Lennon adding a coolly taken second goal before Barkley netted with a deflected shot in the second half.
Seven-time FA Cup winners Chelsea are in action later on Sunday when they visit second-tier Milton Keynes Dons, who humiliated Manchester United 4-0 in last season's League Cup.
There have been no major shocks in the fourth round to date, although West Bromwich Albion were taken to a replay by third-tier Peterborough United after a 2-2 draw on Saturday.
Holders Arsenal, seeking a third consecutive trophy, beat Burnley 2-1 and there were also wins for Manchester United, Manchester City and Tottenham Hotspur, but Liverpool drew 0-0 at home to West Ham United.