Stuart Gray’s side avoided becoming the third Yorkshire side to fall victim to Rochdale when they won 2-1 at Spotland on Saturday.
Second half goals from Joe Mattock and Oguchi Onyewu earned the Owls safe passage.
They will face a Charlton side who made it through thanks to a 1-0 win at Huddersfield Town – former Town loanee Simon Church scoring the only goal in the 54th minute.
Sheffield United were also handed a home tie, should they be able to beat Premier League Fulham in their fourth round replay – a 1-1 draw the outcome at Bramall Lane on Sunday afternoon. If successful at Craven Cottage next week, Nigel Clough’s team will host the winner of the fourth round replay between Nottingham Forest and Preston North End, who drew 0-0 on Friday night.
Hull City – 2-0 winners at Southend United on Saturday thanks to a double from Matty Fryatt – face a tricky trip to Brighton.
Elsewhere, the Premier League’s top four will all face off in the fifth round with Manchester City hosting Chelsea as Arsenal taking on Liverpool.
Manuel Pellegrini’s City, winners in 2011 and runners-up to Wigan last year, recovered from two goals down to beat Watford in the fourth round on Saturday and meet Chelsea following the Blues’ slender 1-0 victory over Stoke.
Ten-time winners Arsenal are currently out in front in the Premier League and tackle Brendan Rodgers’ ambitious Liverpool after the top-flight sides ousted Coventry and Bournemouth respectively.
Other all-Premiership tie see Southampton travel to Sunderland, while Everton host Michael Laudrup’s Swansea. Holders Wigan Athletic travel to Premier League bottom club Cardiff City.
Draw for the fifth round of the FA Cup (matches to be played on the weekend of February 15-16):
Manchester City v Chelsea
Sheffield United or Fulham v Nottingham Forest or Preston
Arsenal v Liverpool
Brighton v Hull
Cardiff v Wigan
Sheffield Wednesday v Charlton
Sunderland v Southampton
Everton v Swansea