The home side's energetic display in the early stages disrupted Chelsea's rhythm going forward, with both Andre Schurrle and Mohamed Salah hitting poor efforts wide as the Premier League leaders were restricted to speculative long-range strikes.
Shrewsbury continued to impress as they attempted to match Jose Mourinho's side, but they were almost caught out by a counter-attack from the visitors until Andre Schurrle decided against heading at goal from Salah's inviting cross.
© Getty Images
A first half which created very few clear chances was soon forgotten as Chelsea took control after the break to pile the pressure on the Shrewsbury defence.
They were rewarded three minutes into the second half when Salah knocked down the ball to Didier Drogba inside the box to allow the Ivorian to fire home the opener.
Schurrle was inches away from doubling Chelsea's advantage moments later with a 25-yard strike which forced Jayson Leutwiler into a fine save.
Leutwiler's night was far from over as he ensured Shrewbury's hopes of causing an upset remained alive by pushing Drogba's powerful shot past the post.
Andy Mangan was sent on in the 75th minute as Shrewsbury went in search of an equaliser, and the substitute made an impact less than two minutes later as he hammered the ball past Cech from close range after Jermaine Grandison had seen his header blocked.
However, the home fans joy was short-lived as one of Mourinho's substitutes helped restored the lead when Willian's dangerous cross was inadvertently headed into the net by Grandison, who was attempting to hold off Drogba in the box.
There would be no late twist as Shrewsbury's admirable bid to cause a shock ended in disappointment at Greenhous Meadow.