Crofts struck just after the half hour although the Seagulls were a touch fortunate as the ball took a crucial deflection off Royals defender Alex Pearce before going in.
It was a stroke of luck the home side deserved after Keith Andrews, Solly March and Ashley Barnes all hit the woodwork before the opener, while Adam Le Fondre wasted a late chance to earn Reading a replay.
Seagulls head coach Oscar Garcia made no few than eight changes and one the call-ups, midfielder Andrews, came desperately close to giving the hosts the lead on two occasions early on.
Andrews forced Australian keeper Adam Federici to tip his deflected shot over in the seventh minute before his fierce drive cannoned back off the crossbar a minute later.
Reading, seeking to avoid a third successive away defeat, continued to be put under early pressure and Jake Forster-Caskey was narrowly off target from the edge of the area after being set up by Andrew Crofts.
Experienced Brighton defender Adam El-Abd did well to thwart striker Le Fondre in front of goal when Gareth McCleary crossed dangerously for Reading to create the Royals first opening.
Only Liverpool and Arsenal have beaten Brighton in 90 minutes in eight previous FA Cup and League Cup clashes at the Amex, and the woodwork again came to Reading's rescue in the 26th minute.
March was unlucky to see his shot come back off the post before Barnes sent the rebound against the bar.
But the Seagulls made their pressure count on 32 minutes when Welsh international Andrew Crofts fired them ahead from just inside the area.
Barnes chested the ball down for Crofts and Reading goalkeeper Federici was beaten as ball took a deflection off Pearce.
Brighton, playing with confidence, threatened again when Kemy Agustien was blocked inside the box following a dangerous run and March later shot over after a strong run.
After a poor showing in the first half, Reading came out with more intent after the break and giant Brighton keeper Peter Brezovan was forced to dive to hold a low effort from Nick Blackman.
Brighton hit back to threaten and, following a pass from Forster-Caskey, March had a lot goal-bound shot at the near post saved low down by Federici.
Australian Federici excelled himself shortly after by producing a flying save to turn a shot from Agustien round the post.
Reading manager Nigel Adkins sent on Dutch midfielder Royston Drenthe in place of Blackman and he was off target with a low shot in the 66th minute.
Brighton, beaten only twice in their previous 13 games, came close to increasing their lead when substitute Lewis Dunk headed a corner from Andrea Orlandi wide 18 minutes from time.
Barnes tried his luck for the Seagulls from 25 yards, shooting wide, before March was withdrawn in favour of Kazenga LuaLua.
Le Fondre wasted a chance to grab a late equaliser for Reading when he wastefully headed a Taylor cross over.
Taylor should also have hit the target in the final minute of normal time, but snatched at his opportunity in firing over.