Rovers had only their second home win of the season sewn up by half-time thanks to two goals from Benni McCarthy and one from strike partner Jason Roberts.
It was the perfect start for new manager Allardyce, whose reign at Newcastle had ended ironically against Stoke with a goalless cup draw at the Britannia Stadium almost 12 months ago.
After 11 league games without a win - a sequence that had ultimately cost Paul Ince his job at Ewood Park earlier in the week - Blackburn were due a break or two and everything went in their favour.
A calf injury picked up by Roque Santa Cruz in training on Friday meant recalls to the starting line-up for McCarthy and Roberts and the two strikers were immediately amongst the goals.
The visitors were left cursing their luck even before kick-off when first choice keeper Thomas Sorensen failed to emerge after the warm-up and his replacement Steve Simonsen was repeatedly left picking the ball out of his net in the first half.
Stoke started well and almost took the lead after a sustained spell of early pressure, but Glenn Whelan's cross struck the top of the bar.
Just two minutes later, Rovers were in front as McCarthy played a clever ball through to Morten Gamst Pedersen, who was sent crashing to the ground by a reckless Ibrahima Sonko challenge and McCarthy slammed home from the spot.
Ten minutes later, Rovers doubled their advantage with only their second attack of the game.
McCarthy teed up Brett Emerton to cross from the right, Ryan Shawcross inadvertently deflected the ball into the path of Roberts, who pounced from close range.
And it wasn't long before the buoyant home fans were celebrating Rovers' third.
McCarthy's fancy flick sent Roberts on his way and the former Grenada international skipped inside Sonko before seeing his shot saved by Simonsen, but McCarthy was on hand to fire home the rebound.
With Chris Samba and Ryan Nelsen winning everything that Stoke threw at them, the visitors were reduced to long shots both before and after the break with Whelan's 56th minute free-kick the closest they came to breaching the Rovers defence.
With a quarter of an hour remaining, Rovers had three chances within a minute to extend their lead further.
With the move of the match, Tugay sent Stephen Warnock scampering down the left, his cross was fired goalwards by Pedersen, but Roberts was offside on the rebound.
Seconds later, Roberts muscled his way clean through only to be denied by Simonsen and from the resultant corner, McCarthy's shot skimmed off the outside of the post.
Stoke had a final opportunity to get on the scoresheet five minutes from time but Paul Robinson made a smart stop from substitute Seyi Olofinjana's vicious volley and Ricardo Fuller dragged his shot inches wide of the far post on the follow-up.