Drogba on target as Chelsea edge it
Jermaine Grandison's own goal broke Shrewsbury's hearts after Chelsea survived a scare to snatch a late 2-1 win.
The Town defender turned in Willian's cross with nine minutes left but Jose Mourinho's side were made to work to reach the League Cup quarter-final.
Andy Mangan had scored 84 seconds after coming on as a 75th-minute replacement to cancel out Didier Drogba's opener before Grandison's unfortunate intervention.
Shrewsbury boss Micky Mellon brought in 17 players in the summer, after only arriving in May himself, with the side made up of homegrown players, free transfers and two loanees.
Only striker James Collins cost a nominal fee from Hibernian.
And front of a record crowd at Greenhous Meadow - 10,210 - the Sky Bet League Two side rattled a Chelsea team containing four Champions League winners and a World Cup champion.
Just Drogba, Gary Cahill, Filipe Luis and Oscar survived from their 1-1 draw at Manchester United while defender Andreas Christensen was given his Blues debut.
James Wesolowski, who had been nursing a groin injury, was replaced at the last minute by Liam Lawrence despite being named on the Shrewsbury team sheet.
The Shrews refused to be overawed early on with Nathaniel Knight-Percival forcing a smart early save out of Petr Cech.
Bobby Grant drove over and Collins headed wide as the hosts initiative, ingenuity and enthusiasm worried Chelsea.
In the middle Ryan Woods dictated from deep while skipper Connor Goldson won his first-half battle with Drogba.
Aside from two weak Andre Schurrle efforts, they contained their visitors and when Chelsea did threaten Schurrle let them off the hook on 32 minutes.
Salah's wild strike then flew out for a throw to underline Chelsea's frustration as they struggled to breach the hosts' organised backline.
When they did, three minutes before the break, Drogba was offside after steering the ball past Leutwiler.
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 csteered past Leutwiler from 12 yards.
The hosts equalised with 13 minutes remaining when substitute Mangan fired in from close range after Chelsea failed to clear Lawrence's corner.
But they were level for just four minutes when Grandison turned Willian's low cross into his own net to end home hopes.