Dembele bagged the first hat-trick of his professional career, assisted on one other goal and was involved in the incident that saw Shrewsbury midfielder Josh Vela sent off in first-half stoppage time.
The 24 year-old takes some terrible punishment from less talented opponents, but nothing could put him off his stride today.
Jonson Clarke-Harris, from the penalty spot, and Jack Taylor, from the edge of the area, scored the goals that bookended Dembele’s super show.
Shrewsbury had briefly drawn level in the first-half, but were behind again when reduced to 10 men before getting overrun in the final quarter of the game.
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Posh returned to their wing-back formation for this game with Niall Mason replacing Nathan Thompson in the starting line-up.
But Posh started poorly and required two fine defensive blocks early in the game to avoid falling behind from set-pieces. Centre-backs Ethan Ebanks-Landell and Aaron Pierre were the unfortunate Shrews players with Frankie Kent and Dan Butler responsible for the blocking.
Posh then sprung into the life and took the lead in the 16th minute. Dembele caused the problems racing onto Kent’s long pass and nipping in to beat Shrews ‘keeper Harry Burgoyne to the ball. Burgoyne’s response was to flatten the Posh striker and Clarke-Harris stepped up to score his fourth goal in four matches - two of them from the penalty spot.
Posh weren’t able to enjoy the moment for long though as within 60 seconds little Josh Daniels was nodding home Marc Pugh’s fine left wing cross. ‘How did someone that small win that header,” roared Posh goalkeeper Christy Pym as the visitors celebrated just a sixth goal of the season.
Posh did improve their passing as the half wore on. Dembele smacked over the bar after good play from Reece Brown and Joe Ward. Ward then cleared the crossbar with a 25 yard free kick.
Posh were unafraid to use the long pass and a second goal arrived from that tactic two minutes before the break.
Sammie Szmodics was the supplier to Dembele who was left one-on-one with Pierre in the penalty area. The result was inevitable as Dembele skipped past the centre back before slotting home with ease.
And things got much worse for injury-plagued Shrewsbury in the second minute of added time as Vela’s crude lunge on Dembele led to an instant red card.
Shrewsbury’s response was to bring on former Posh striker Jason Cummings, while Darren Ferguson sent his side out with a flat back four.
And Posh should have moved two goals clear, again after superb work from Dembele who drew the attention of two defenders before feeding Szmodics who was unmarked 15 yards from goal. His shot never threatened the goal.
Initially Posh struggled to dominate 10 men. Josh Daniels fired wide from 25 yards and there was an awkward moment when Pym caught the ball under his bar and collided with an upright. Luckily he didn’t drop anything.
Charlie Daniels and Josh Daniels both sent shots wide from distance, but the Shrews played very open for a side with 10 men and the inevitable happened 20 minutes from time.
Szmodics picked the ball up in space in the centre circle, freed Ward expertly down the right and wing and League One’s assist king played the perfect square ball that Dembele converted with ease.
Szmodics was excellent in possession, but struggled to find a finish, firing well wide after bursting into the penalty area and then failing to beat Burgoyne from close range after some neat work from substitute Mo Eisa. Eisa was making his first appearance in a Football League match since the second game of the season.
Dembele completed his hat-trick when accepting Butler’s fine through ball, dancing around Burgoyne and slotting into an empty net 10 minutes from time.
There was still time for the unfortunate Szmodics to be defied again by Burgoyne when he met substitute Idria Kanu’s cross from close range with his knee and the goalkeeper saved well from Butler before Jack Taylor struck the fifth and final goal from 20 yards after decent play from Eisa two minutes from time.
The win was big enough to send Posh top of three teams, Ipswich and Lincoln are the others, on 22 points after winning six of seven unbeaten games in October.
Posh: Christy Pym, Niall Mason, Dan Butler, Mark Beevers, Frankie Kent, Jack Taylor, Reece Brown (sub Louis Reed, 53 mins), Joe Ward (sub Idris Kanu, 82 mins), Sammie Szmodics, Siriki Dembele, Jonson Clarke-Harris (sub Mo Eisa, 74 mins). Unused subs: Dan Gyollai, Ryan Broom, Frazer Blake-Tracy, Ethan Hamilton.
Shrewsbury: Harry Burgoyne, Ro Shaun-Williams, Aaron Pierre, Ethan Ebanks-Landell, Charlie Daniels, Oliver Norburn, Danuel Udoh (sub Shillo-Tracy, 81 mins), Scott High, Marc Pugh (sub Jason Cummings, 46 mins), Josh Daniels.
Unused subs: Cameron Gregory, Jan Zamburek, Brad Walker, Marlon Fossey, Ryan Bennett.
Goals: Posh - Clarke-Harris (pen, 17 mins), Dembele (42 mins, 69 mins & 80 mins), Taylor (88 mins).
Shrews - J. Daniels (18 mins).
Sending-off: Shrews -Vela (serious foul play).
Cautions: Posh - Brown (delaying the restart), Szmodics (foul).
Shrews - Burgoyne (foul), Norburn (foul), Ebanks-Landell (foul)
Referee: James Oldham 8