Peter Crouch's third goal in as many games had put Mark Hughes' side ahead early on in a scrappy first half, with Blackburn failing to stifle Stoke's forward line.
However, despite riding their luck at times, the hosts regrouped to complete a fantastic first-half turnaround.
King, replacing Jordan Rhodes in the starting line-up, levelled with a smart header before Rudy Gestede put them ahead with a coolly taken penalty after Geoff Cameron had been sent off for bringing the striker down in the area.
Norway international King's quick-fire double early in the second half put the result beyond all doubt, as Hughes suffered a second consecutive humiliating defeat against a former club and the Championship outfit sailed into the quarter-finals.
Hughes, who guided Blackburn to an FA Cup semi-final during his four-year managerial stint at Ewood Park, made just one change to the Stoke line-up that lost to Manchester City last time out, with Jack Butland replacing Asmir Begovic in goal.
And it was the visitors who took the initiative when, in the 10th minute, Blackburn failed to clear a Victor Moses corner and Crouch was on hand to prod home from close range.
Blackburn could have levelled 10 minutes later as King picked up possession just inside the area, but his venomous effort clipped the top of the crossbar.
Stoke should have been further ahead after goalkeeper Simon Eastwood parried Moses' effort into the path of former Blackburn loanee Mame Biram Diouf, but the Senegal forward blasted wide from eight yards out.
Hughes' side were made to pay for that miss when Shane Duffy headed goalwards from Craig Conway's corner and King leapt up in front of Butland to nod it in from a yard out.
King almost turned provider when he raced in behind the Stoke backline and slid a perfect cross into the feet of Gestede, but Blackburn's top scorer failed to convert.
The forward atoned on the stroke of half-time by sending Butland the wrong way from the penalty spot after Cameron had brought him down on the edge of the six-yard box – with the American seeing red as a result.
Blackburn made the numerical advantage count five minutes after the restart as King once again split open Stoke's defence and, after Marc Muniesa slumped to the ground with an apparent hamstring injury, King raced through to score past Butland.
King grabbed a third six minutes later, when, in almost identical circumstances to his second, the 23-year-old latched onto Getsede's pass before firing calmly into the bottom-right corner.
Gestede had a goal ruled out for a foul on Butland in injury time, while trouble in the away end added to Stoke's miserable day in Lancashire.
