Blackburn carried the greater threat against their Premier League opponents in the first half, but they were unable to take advantage of the openings they created.
Eddie Howe's side subsequently broke the deadlock in the 71st minute when Anthony Gordon netted his first-ever goal in the FA Cup.
As it proved, Rovers were behind for just eight minutes before Sammie Szmodics kept his composure from a rebound to send an enthralling tie to extra time.
With neither side able to find a winner in the additional 30 minutes, the fifth-round tie was decided by a penalty shootout, as Martin Dubravka made two decisive stops to send Newcastle through to the quarter-finals for the first time since 2019-20.
Newcastle travelled to Ewood Park looking for a positive response after falling to a heavy 4-1 defeat in Saturday's away meeting with Arsenal.
However, it was the hosts rather than the Magpies who carved out the most promising chance of the opening half-hour, with Szmodics firing into the side-netting after being teed up by John Buckley.
Blackburn then threatened through Tyrhys Dolan, who saw his driven strike pushed behind by the right hand of Dubravka.
After being put on the back foot by their Championship hosts, Newcastle offered their own threat in the 33rd minute when Sean Longstaff saw his near-post strike kept out by Aynsley Pears.
The dangerous Szmodics created one final chance of the first period, working the space to unleash a fierce left-foot strike which forced Dubravka to tip over the crossbar.
Rovers built on the momentum built from their encouraging first-half performance to make a positive start to the second period, with Buckley drawing a smart save from the outstretched Dubravka.
Newcastle began to grow into the second half as Jamaal Lascelles towered high from Kieran Trippier's corner, only to see his header kept out on the line by Kyle McFadzean.
Gordon was then presented with a glorious one-on-one opportunity, but the forward could only guide his strike straight at the resolute Pears.
The 23-year-old was only made to rue that spurned chance for a couple of minutes, before he swept the ball home from Miguel Almiron's neat cut-back.
Gordon's ninth goal of the season did little to deter Blackburn, who pulled themselves level after Lascelles was punished for missing a routine header from a throw-in.
Rovers quickly worked the ball to Markanday, and although the 22-year-old saw his effort rattle the frame of the goal, Szmodics was on hand to turn home the rebound.
With momentum in their favour, the hosts went agonisingly close to snatching a winner in regulation time when Yasin Ayari's deflected strike was kept out by Dubravka.
As the contest moved into an additional 30 minutes, the Magpies began to up the ante, with Pears forced to make two sensational saves in quick succession to keep Gordon and Langstaff at bay.
Harvey Barnes then saw his header kept out by the determined Blackburn goalkeeper, before Scott Wharton made a crucial block to prevent Bruno Guimaraes from winning the tie.
There was still time left for Dubravka to pull off a critical stop to thwart Szmodics and ensure a dreaded penalty shootout would be required to find a winner.
Although Barnes failed to convert his spot-kick, Dubravka made two critical saves to deny Szmodics and Dominic Hyam, ensuring Newcastle will be involved in Wednesday's quarter-final draw.