Blackburn were handed the perfect start, taking the lead at The Valley after just four minutes as Taylor climbed highest to head home Craig Conway’s cross.
After a brief delay for floodlight failure, Johann Berg Gudmundsson responded for the hosts direct from a free kick after his left-footed shot found the bottom corner.
However, Taylor responded straight away as his right-footed effort from the centre of the box beat Neil Etheridge to give the advantage back to the visitors.
Charlton searched for an equaliser to force a replay but their chances were dashed when they were reduced to 10 men after Yoni Buyens picked up two bookable offences as Blackburn held on for their fourth win in six visits to The Valley.
Rhodes scored both goals in Blackburn’s 2-0 win at Ewood Park when these sides met a fortnight ago and Chris Brown started up front in his absence.
Rovers scored with their first shot after just four minutes as Conway crossed from the left for Taylor to rise highest in the area and plant a header into the bottom left corner from around eight yards.
Charlton were without a win in seven league matches coming into this game and looked a side short on confidence, particularly in the absence of suspended winger Callum Harriott.
They looked fairly menacing down their right flank, as Blackburn did down their left, and gained their only real chance of the opening period as George Tucudean crossed from the right and the ball fell to captain Johnnie Jackson, who could only fire over from 12 yards under pressure from Ben Marshall.
Five minutes after the game resumed from the floodlight failure, Blackburn could have had a second goal as another Conway cross from the left was headed straight at Charlton goalkeeper Etheridge by Brown.
However, Charlton were level 10 minutes after half-time as Gudmundsson curled a brilliant free-kick over the wall and low into the bottom-right corner from around 25 yards.
But parity lasted for just another four minutes as Brown did brilliantly to turn Charlton right-back Lawrie Wilson one way and then another before crossing for Taylor to blast his second of the game into the roof of the net.
Charlton’s hopes of getting back into the game were effectively extinguished as Buyens was dismissed for his second booking on 72 minutes, while Blackburn’s Tom Cairney rattled the bar from just outside the area with a brilliant drive near the end.