The Royals’ decision to rest a number of players on Saturday looked a wise one as they raced into a 2-0 lead inside the opening 10 minutes against the Bantams.
Hal Robson-Kanu headed them in front in the sixth minute and Garath McCleary doubled the advantage shortly afterwards with his first goal of the season.
Bradford had come from 2-0 down to stun Chelsea 4-2 in the fourth round but were reduced to 10 men after 63 minutes as Filipe Morais was sent off for a tackle on Nathaniel Chalobah.
Jamie Mackie then put the result beyond doubt, and set up a meeting with Arsenal at Wembley, as he made it 3-0 five minutes later.
The Royals will be appearing in the last four of the competition for only the second time in their history after ending Bradford’s fairytale FA Cup run.
The League One side had knocked out two Premier League sides on their way to the quarter-finals but were soon up against it at the Madejski as they fell 2-0 behind.
Reading, who made 11 changes from the side that lost 4-1 to Watford in the Championship on Saturday, went ahead with the first chance of the match as Robson-Kanu’s headed in from McCleary’s corner.
McCleary was tormenting Bradford and he put the tie firmly in Reading’s hands as he cut inside before hitting a shot that ballooned off of Andy Halliday and looped over Ben Williams to double the hosts’ lead.
Danny Williams went close to making it 3-0 with a shot that the keeper pushed away while McCleary also fired over from a good position.
Federici was called into action on the stroke of half-time as Billy Knott could not make the best of a Bradford breakaway.
The second half started without the same fanfare as the first as Reading looked to complete a professional job and Bradford toiled to get a foothold on the game.
The first incident of note came when Morais was dismissed for catching Chalobah with a high foot as the two looked to win a long ball – with referee Mike Jones originally pulling out a yellow card before deciding the former Chelsea youngster’s challenge was worthy of a straight red.
The game was then killed off by Mackie, who timed his run to perfection to latch onto Chalobah’s through ball and thrash an effort past Williams to send Reading into only the second FA Cup semi-final in the club’s 144-year history, with the home fans running on to celebrate on the pitch at full-time.