Goals from Leroy Lita and Shane Long in the first half had the Royals two up at half time.
A thumping header from Sam Sodje made it 3 - 0 before Ade Akinbiyi pulled one back for the Claret's on 69 mins.
Garreth O'Connor's 90th minute goal made it 3 - 2 but it was too late for Burnley to push on for an equaliser.
Reading now face a trip to either Newcastle or Birmingham City in round 4.