In front of a record breaking crowd of more than 10,000 fans at Greenhous Meadow, a Didier Drogba strike and a Jermaine Grandison own goal saw the Premier League leaders to a 2-1 victory.
But Shrewsbury didn't go quietly with Andy Mangan pulling Town level with just over 10 minutes to go when he fired home from close range following a Liam Lawrence corner.
Sadly for Town it just wasn't enough as Grandison then sliced into his own net moments later while under pressure from Drogba.
Having won their last four games, Mellon initially kept faith with the same team that secured the 2-1 victory against Portsmouth on Saturday.
However, after picking up a slight groin injury in that game, James Wesolowski aggravated the problem in the warm up and was replaced by captain Lawrence just before kick-off
Jean-Louis Akpa Akpro retained his place up front with Mellon sticking with the striker who scored a hat-trick against Bury.
It meant Andy Mangan was forced to make do with a place on the bench.
More interestingly was Mourinho's team selection with the Portuguese manager claiming his side were "in trouble" before kick-off due to the amount of injuries in the squad.
Despite playing Manchester United at Old Trafford on Sunday, the self-proclaimed Special One still named an incredibly strong line-up which included Stamford Bridge legend Didier Drogba.
Also in the side was world cup winner Andre Schurrle, Brazilian international Oscar and England vice-captain Gary Cahill.
Meanwhile, the Chelsea bench was packed with talent costing tens of millions of pounds with first team regulars Eden Hazard, Nemanja Matic and Willian all named amongst the substitutes.
Chelsea settled into the game quickly and won a succession of early corners but Shrewsbury showed they were up for the fight by creating the game's first real chance.
A quick break led to Town winning a corner of their own and Lawrence's delivery was met by the head of Connor Goldson with Petr Cech forced to save on the goal line.
Buoyed, Town burst forward once more and a Goldson chip forward was chased by Akpa Akpro who did brilliantly to beat Cahill to the ball.
The French striker fizzed a low cross in front of the six yard box with Cech getting down well to smoother quickly.
Chelsea had been reduced to two long distance efforts by Schurrle in the first 10 minutes but otherwise Town had settled well.
And Mellon's side continued to show their ambition when Woods found Grant with an incisive pass.
The on-loan Blackpool midfielder turned and ran towards goal before hitting a rising shot which sailed just over the bar.
Sensing they were in a game, Schurrle tried to slip in Drogba by the penalty spot but Goldson was on hand to slide in and clear the danger.
Undeterred, the men in blue and amber burst forward again. Grant fizzed a lovely ball wide left to Mickey Demetriou whose cross was met by the head of James Collins.
Sadly for the Town striker, this time his effort sailed wide of the post with the striker unable to reproduce the goal scoring heroics he showed against Norwich in the last round.
As the game hit the 20 minute make Chelsea's Nathan Ake picked up the game's first booking for a late challenge on Grant who was really stamping his authority on the game.
And moments later Town almost took the lead again when Akpa Akpro crossed to James Collins who took a touch and tried to spin Cahill. However, the England vice-captain stayed strong to muscle him away.
With Woods continuing his impressive league form in central midfield, the Meadow was rocking with loud chants of "Salop" and long choruses "Oh When the Town Go Marching In."
But it was Chelsea who nearly took the lead following a quick break from a Town corner.
Drogba did brilliantly to control the ball in his own box and dribble away before launching Mohamed Salah on a swift counter.
The Egyptian sped 30 yards and then clipped the ball to the unmarked Schurrle who inexplicably headed across goal rather than just nestling the ball past Jayson Leutwiler in the Town goal.
Chelsea were continuing to lack any real conviction, with Oscar shooting over, then Mikel lost the ball to Grant much to the frustration of Mourinho who walked back to his dugout and called for Willian to warm up.
To make matters worse for the Special One, Salah then hit a shot from 30 yards which went so far wide it sailed out for a throw in.
Just before the referee blew for half-time, Drogba got clear of the Shrewsbury backline and fired pass Leutwiler but the linesman had his flag up with the striker comfortably offside.
Shrewsbury were cheered off at the break - and deservedly so following an excellent performance that had seen them close down all over the pitch and defend from the front while also looking comfortable on the ball.
Town began the second half kicking towards the Salop Leisure stand but it was Chelsea who started the brighter with Oscar seeing a low strike diverted wide for a corner.
And moments later the men from Stamford Bridge took the lead through Drogba.
The striker, who has played at the Greenhous Meadow before in a pre-season friendly for Galatasaray, ghosted into the box and played a one-two with Salah before slotting the ball past Leutwiler from eight yards.
But Shrewsbury continued to believe in their own style and on 55 minutes a long Demetriou throw was cleared to Akpa Akrpo on the edge of the box.
The French striker took a touch and fired a low shot towards goal which was deflected just wide with Cech rooted to his goal line.
Collins then fizzed a low drive just past the Chelsea 'keeper's right hand post as Town looked to fight back.
At the other end, Leutwiler did brilliantly to dive at full stretch and tip a Drogba piledriver over for a corner.
As the game ticked past the hour mark, Town again came close to getting back on terms.
A Liam Lawrence corner was met by the rising Jermaine Grandison but his header was comfortably claimed by Cech.
Mellon made his first change replacing Grant, who had been less effective since then interval, with youngster Jordan Clark.
Chelsea taking the lead had stifled the game but Mellon decided to throw the dice once more, replacing Collins with Mangan.
And seconds later there were wild scenes around the Meadow when Mangan scored with his first touch after turning in from close range after Lawrence's corner.
It was a scrappy goal but nobody in the ground cared with Mangan running off to celebrate in pure jubilation as 10,000 screaming Salopians jumped for joy.
But the party was to be short lived with Town soon finding themselves behind again in the cruellest of circumstances.
Substitute Willian's cross was sent into the box and as Drogba threatened, Grandison stuck out a leg and unlucky diverted the ball into his own net with just 10 minutes remaining.
It was a harsh blow for Town and Grandison, who up until that point had been superb.
Shrewsbury immediately tried to hit-back and a deep Grimmer cross was met by Akpa Akpro, who headed just over the bar.
But Chelsea saw out the remaining five minutes comfortably, with Micky Mellon's side loudly clapped and cheered by the record breaking Meadow crowd at the final whistle.
And the adulation was richly deserved as Town performed heroically against the Premier League leaders who are 72 places above them in the football pyramid.

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