Rovers were the better side in the first half and deservedly led at the interval through Gael Givet's stunning strike.
But Heskey levelled matters with his first touch after 59 minutes and then Young struck twice in quick succession to see Villa - last season's beaten finalists - reach the last 16.
Rovers must be sick of the sight of Villa after being knocked out of both domestic cup competitions by them last season.
Villa looked out of sorts in the first 45 minutes as Houllier opted for a 4-3-3 formation with John Carew flanked by Gabriel Agbonlahor and Young.
But they were transformed in the second half with Young in inspired form but midfielder Barry Bannan and defender Carlos Cuellar also made key contributions.
Villa Park was under half full for Houllier's first match at the helm as he made eight changes from the side which faced Bolton at the weekend.
Young forced the first save of the game out of Mark Bunn after turning past Morten Gamst Pedersen but the Rovers goalkeeper held onto his low 25-yard drive.
Rovers retaliated and Brad Guzan beat out Phil Jones' rising drive from a narrow angle after being found in space by Pedersen.
Villa midfielder Steve Sidwell was just off target with a long-range drive after being teed up by Bannan.
Villa suffered a setback after 31 minutes when Agbonlahor limped out of the action to be replaced by England Under-21 Nathan Delfouneso.
The England striker has been dogged by injury niggles during the early part of the campaign and he went straight to the dressing room for treatment.
Then after 34 minutes Rovers stunned Villa by taking the lead through Givet's powerful strike.
Pedersen's throw was headed out by Ciaran Clark to Givet whose powerful drive flew through a crowd of players into the corner of the net.
Blackburn were finding plenty of space and Steven Nzonzi was allowed to move forward unchallenged but his shot was wide of the mark.
Carew was brought down by Nelsen just outside the box in first half injury-time but Young curled the free-kick over the crossbar.
It needed a fine save from Guzan to prevent Blackburn from doubling their lead early in the second period.
Pedersen's long throw was cleared back to him and the resulting curling low shot from the edge of the box was turned aside by the United States goalkeeper.
Houllier brought on Emile Heskey, whom he signed for Liverpool from Leicester a decade ago, in place of Carew after 58 minutes.
It proved a masterstroke move with Heskey bringing Villa level with his first touch of the ball inside 60 seconds.
Young sent the former England player clear of the Rovers defence and he clinically beat Bunn with a low drive.
Villa were now in the ascendancy and Bunn finger-tipped away a powerful Bannan drive.
Then Cuellar should have done better when he headed wide when well placed from a Young corner.
But with 14 minutes left, Young put Villa ahead.
Delfouneso supplied the near-post cross and Young was able to head past Bunn.
Before the restart the injured Villa defender Ciaran Clark was replaced by Curtis Davies.
But within 60 seconds Young doubled Villa's lead.
Heskey won the ball on the right and surged into the Rovers box before delivering an inch-perfect cross which Young converted from close range.
Young came close to completing a hat-trick when his 20-yard drive was only just too high.