In a close-run encounter at a chilly Ewood Park, nothing could separate the two sides over 120 minutes of football, meaning it would need penalties to decide who would face minnows Broxbourne Borough in the fifth round.
12 spot-kicks were converted successfully, but after Louis Hall also found the back of the net for the visitors, Joel Steer was the unfortunate man to miss as he put his effort over the bar to ensure it would be the visitors who would go through.
The first half was a tight-knit affair with very little between the two evenly matched sides.
The visitors were the first to have sight of goal when defender James Finnerty headed an early corner over, whilst at the other end Jack Evans fired wide as play opened up for the midfielder.
The best chance of the opening 45 for Villa came just before the midway point of the half, and again it was from a set-piece. A corner from Jake Doyle-Hayes picked out Finnerty once more, but whilst the defender met the ball with his head powerfully, he was unable to direct it on target.
Rovers improved as the half wore on with Tre Pemberton seeing a shot deflect wide after a fine passing move. Then they too would go close with a headed opportunity from a corner kick on 35 minutes.
Joe Rankin-Costello's in-swinger found Matty Makinson who headed goalwards, but Villa keeper Victor Johansson pulled off a fantastic fingertip save to prevent the ball from finding the back of the net.
Evans would try his luck from distance as the first half drew to a close, but like his earlier effort it didn't trouble the Villa goal.
Stefan Mols headed an early chance wide at the start of the second half, before Harvey Knibbs did likewise for the visitors.
But in a half littered with petty fouls, flowing football was hard to come by, but Charley Doyle did force Johansson into a near post save after he turned well in the area.
However the midfielder would go closer with yet another headed chance from a set-play. Rankin-Costello picking out Doyle with his in-swinging corner, but he glanced inches wide.
The visitors fashioned a late chance with Jake Doyle-Hayes shooting from distance, but as it cleared Ben Ascroft's bar it meant 30 minutes of extra-time would be required.
But just like the 90 minutes that had gone before, chances were few and far between when play resumed in extra-time.
Makinson saw a volley blocked from a Joe Grayson corner. Before Ascroft was called upon to make his first save of the night on 103 minutes, getting down well to his right to keep out a Jordan Cox strike.
Then in the second period sub Joel Steer did well on the left to feed Rankin-Costello, but his centre was put behind by Doyle-Hayes with Mols ready to pounce.
The resulting corner saw Grayson pick out Rankin-Costello at the near post, but he glanced the ball wide. It would prove to be the last chance of the game meaning spot-kicks would be required to decide who would go through to the fifth round.
The first 12 penalty kicks were all converted, but after Louis Hall had netted for the visitors to make it 7-6, Joel Steer was the unfortunate Rover to miss as he put his effort from 12-yards over the bar.
ROVERS: Ascroft, Jackson, Thompson, C. Doyle, Makinson, Magloire (c), Pemberton, Mols, Butterworth (sub Steer 79 mins), Rankin-Costello, Evans (sub Grayson 72 mins).
Subs not used: Campbell (GK), Wright, Lynch.
Booked: Jackson 110 mins.
ASTON VILLA: Johansson, Walker, Hall, Prosser, Finnerty, Clark, Knibbs (sub Rowe 82 mins), Doyle-Hayes (c), Mooney, Clarke, Cox.
Subs not used: Williams, Idem (GK), Abdoul, Coates.