The hosts were first to everything in the first half, and the gulf between the pair looked non-existent as the Shrews more than held their own.
Micky Mellon’s men restricted their illustrious guests to half chances and long range shots, with Shrews’ stopper Jayson Leutwiler untroubled.
However, immediately after the interval, the Premier League leaders showed their class, with an incisive move leading to the opener for Didier Drogba.
The hosts toiled in awful conditions, but for all their endeavour, the Shropshire side could not find a way through a resolute Chelsea rearguard.
However, just 90 seconds after coming off the bench Andy Mangan equalised from close range to send the home fans into raptures.
That delirium was short lived however, as one of Jose Mourinho’s own substitutes Willian crossed into the danger area, only for Shrews defender Jermaine Grandison to nod past his own ‘keeper to give the Blues the lead again just four minutes later.
Chelsea hung on to secure a place in the quarter-finals, while the Shrews head out with their heads held high after a fantastic effort.
With the rain pouring down the Shropshire side set their stall out early on, and didn’t allow the visitors to settle on the ball.
The Shrews actually had the chance of the half as early as sixth minute, with Petr Cech forced into a smart save after Nathaniel Knight-Percival got up highest to head a corner through a crowded penalty area towards goal.
Chelsea, who made seven changes from the weekend draw at Manchester United were very flat going forward in the first half, with Mohammed Salah the only really threatening Blues player.
The Egyptian was allowed to cut inside on several occasions, but he failed to hit the target with Shrews’ ‘keeper Jayson Leutwiler untroubled.
Andre Schurrle did have a great chance at the back post after being picked out by Salah, but the World Cup winner’s header was mistimed and the danger was cleared.
There had been little to concern Shrewsbury in the first half but within three minutes of the re-start they were behind.
Schurrle and Salah combined well to tee up Drogba who clinically steered past Leutwiler from 12 yards.
It was the first goal Shrewsbury had conceded in the competition but they almost levelled eight minutes later.
Lawrence’s free-kick was knocked down to Goldson and his shot deflected off Collins to fly wide with Cech helpless.
Collins and Drogba traded efforts from distance but Town began to run out of steam with Chelsea content.
Yet the hosts did equalise with 13 minutes remaining when substitute Mangan, on for Collins, fired in from close range after Chelsea failed to clear Lawrence’s corner, with Mikel caught flat-footed as he attempted to clear.
But they were level for just four minutes when Grandison turned Willian’s low cross into his own net to end home hopes, but not take awya what a valiant effort it was from the League Two promotion hopefuls.