It's Newcastle United that will take on Watford in the fourth round of the Emirates FA Cup, but it needed extra time for the Magpies to edge past a resilient Rovers.
In a lively game, a couple of extra time goals from Joselu and substitute Ayoze Perez won the tie for the visitors after Rovers had initially recovered from going 2-0 down early in the first half.
Young guns Sean Longstaff and then Callum Roberts put the Tynesiders in command at Ewood Park, but goals from Adam Armstrong and Darragh Lenihan levelled things up just before half time.
The turning point in the game appeared to come when substitute Bradley Dack skewed a strike wide before Joselu made it 3-2, with Perez making the tie safe later on in extra time.
Rovers got off to the worst possible start as Longstaff's strike took a deflection and flew into the back of the net after just 56 seconds at Ewood Park.
Rovers did appear to settle into the game, but it was the visitors who had the next opportunity of the game as Matt Ritchie forced David Raya to save with his legs on 13 minutes.
Smallwood was the first Rover to have a shot in anger, but whilst Longstaff got lucky earlier in the game, the returning number 6's strike was deflected wide of Freddie Woodman's goal.
But two minutes later and the Magpies doubled their advantage through Roberts, who latched onto a low cross from Jacob Murphy to whip the ball first time past Raya at the near post.
However, just after the half hour mark Rovers were given a lifeline. Cute play from Danny Graham saw one Geordie feed another as Armstrong latched onto his pass to prod into the back of the net against his old club, despite Woodman getting a tough on the effort.
That was just the tonic for the hosts and a Harrison Reed corner was narrowly headed over by Lenihan, as Mowbray's men searched for a quickfire equaliser.
But that leveller did finally come, as Reed found Lenihan again with an excellent corner, and this time he sent a thumping header past Woodman and in on the stroke of half time.
It was probably what Rovers deserved having had more of the ball in the final 10 minutes of the half in particular.
Newcastle made a change at the break with club captain Jamaal Lascelles replacing Ciaran Clark in a straight swap from Rafa Benitez.
But Lascelles' evening lasted just 10 minutes, as the central defender limped off with a hamstring injury. Isaac Hayden came on to replace the former Nottingham Forest man, with Fabian Schar moving from holding midfield to the heart of defence.
Hayden almost made an immediate impact when he cracked the post with a header after latching onto Ritchie's outswinging corner kick with 22 minutes remaining at the visitors looked to regain the lead.
But neither side could find a winner in the 90 minute period and the tie would go to extra time to book a place in round four.
Rovers continued to probe in extra time, with substitute Joe Nuttall glancing Amari'i Bell's cross just past Woodman's right-hand post.
Right on half time of extra time Dack had a terrific opportunity to hand Rovers the advantage, but he could only fire wide after Craig Conway's intelligent flick laid it into his path.
A moment later and Newcastle regained the lead through Joselu, who tucked past Raya after the goalkeeper parried Ritchie's swerving free kick into his path. The striker was offside, but without VAR, referee Lee Probert was only given one look.
Rovers had to go for it at the beginning of the second half of extra time, and they were caught out as a long ball found substitute Ayoze Perez, who beat Lewis Travis for pace and fired past Raya.
It was a bitter blow for Rovers, who would have been planning a big finish to what was a pulsating game.
The lads didn't give up, but Raya did well to use his legs again to deny Joselu late on before Christian Atsu thumped wide of the target.
Rovers will count themselves unfortunate to bow out of the competition, and will be hoping for better luck when Ipswich Town visit Ewood on Saturday, kick-off 3pm.
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Rovers: Raya, Bennett (c), Nyambe, Lenihan, Bell, Travis, Smallwood (Rothwell, 102), Reed (Dack, 78), Brereton (Conway, 65), Armstrong (Nuttall, 73), Graham.
Unused substitutes: Leutwiler, Grayson, Butterworth.
Goals: Armstrong, 33, Lenihan, 45
Newcastle United: Woodman, Clark (c) (Lascelles, 46 (Hayden, 56)), Schar, Murphy, Ritchie (Atsu, 118), Fernandez, Manquillo, Joselu, Sterry, Longstaff, Roberts (Perez, 94).
Unused substitutes: Harker, Rondon, Yedlin.
Goals: Longstaff, 1, Roberts, 22, Joselu, 115, Perez, 117.
Booked: Fernandez, 90
Referee: Lee Probert.