Nicky Adams rocked the Premier League side when he scored his ninth goal of the season inside the first minute.
Reds could have stretched their lead before Adam Le Fondre levelled on 13 minutes.
Noel Hunt got the second goal for the visitors just before the break and a controversial penalty, awarded for a foul on Hunt by Joe Walsh, made it 3-1 early in the second half.
Reds kept plugging away but it's Reading who go through to the fourth round.
Richie Barker named an unchanged side and Reds tore into their opponents and were ahead straight from the kick-off. Adams collected a diagonal pass from Mark Connolly wide on the right and as the visitors backed off Adams hit a superb shot from 20 yards into the top right-hand corner.
It could have got even better for the home side as they continued to make the early running. Adams went close again, this time with his right foot, and on nine minutes Royals' 'keeper Adam Federici tipped over a header from Joe Walsh at full stretch.
But on 13 minutes Reading levelled against the run of play. Gareth McCleary cleverly headed the ball back into the path of Le Fondre who picked his spot from 15 yards.
It was a body blow to Reds and their fans in a sell-out crowd of 5,880 but they continued to threaten. Josh Simpson tested Federici who then made a sprawling save at the feet of Mke Jones.
So it was a surprise when Reading scored against the run of play for the second time just before the break. A slip by Mat Sadler enabled McCleary to find Hunt who again gave Paul Jones no chance with his close-range finish.
The crucial goal came four minutes after the break when Walsh was adjudged to have brought down Hunt. The defender claimed he got the ball and its trajectory after the incident supported his claims. There were few appeals from the Reading players but the referee pointed to the spot and Le Fondre lashed the penalty into the roof of the net.
Reds refused to go quietly. Billy Clarke's cross just eluded Gary Alexander and then Adams and Clarke both forced Federici into saves. Reading were a threat on the break but it would have been extremely harsh had they conceded again.
Lateef Alford-Alliyu came on for his debut late on and showed some promising touches but the FA Cup dream is over for another year for Reds.