At that stage it appeared to be plain sailing for the Premiership side, but from the second half onwards the Baggies crumbled.
A wicked shot on the turn on 50 minutes was followed by a superb second by Lita to set up an additional 30 minutes.
The 21-year-old hitman completed his treble, and the fightback, when he swept home three minutes into extra-time.
The last thing either side wanted was extra-time, though for very different reasons.
West Brom, struggling near the Premiership drop zone, looked to have won the first game at The Hawthorns with a late penalty from Zoltan Gera only for Reading to force a replay with Kevin Doyle's late spot-kick.
And the replay proved an extended distraction for both sides, with Reading particularly wary of another two hours of football ahead of their Friday night Championship clash with Crystal Palace.
So when Chaplow opened the scoring after nine minutes, neatly finishing into the far corner after Ronnie Wallwork had played him clear of a muddled defence, it looked like the Premiership side had taken the sting out of the tie.
And it appeared to be all over on 32 minutes when Chaplow, having gone 12 months without a goal since his move from Burnley, tapped home the second after Reading keeper Graham Stack had fumbled Nathan Ellington's snap shot.
But Reading, unbeaten in the League since the opening day of the season, seem to have forgotten how to lose.
Five minutes into the second half they began their fightback when John Oster played Lita into space and he controlled with his right foot, before turning and smashing home a shot with his left.
A minute later Chaplow's back pass was handled by West Brom goalkeeper Chris Kirkland, but the error went unpunished.
It was, however, a brief reprieve.
Midway through the second half Lita, making his first start since December 3 following an ankle injury, swept a superb shot over the head of Kirkland after a stunning Reading build up inspired by the impressive James Harper.
Three minutes into extra-time the comeback was complete. Steven Sidwell appeared to be fouled on the edge of the box but Reading were allowed to play the advantage, enabling Harper to bundle the ball into the path of Lita, who made no mistake.
The delirious striker was booked for ripping off his shirt in celebration but it was a small price to pay for a remarkable fightback and a dazzling hat-trick.