The 1-0 reverse, a first home league defeat in seven dating back to October 24, was hard on Town who were at least worthy of a share of the spoils on a night where they were on top for large spells.
But the hosts were punished for poor defending as Owen Dale notched the game’s only goal from close range on 27 minutes.
Town, under Aaron Wilbraham and David Longwell as Cotterill continues to recover from Covid-19 away from the club, were unable to build on a promising victory against Peterborough on Saturday despite a number of efforts on goal, as the visitors defended well.
Shrews, who remain 17th, failed to carve a better chance than the one Harry Chapman missed inside a minute and the visitors enjoyed a third league win on the spin in Shropshire.
Newtown youngster Ryan Sears, who had been a part of the Town matchday squad each of the three fixtures since the return from the Covid-19 break, was handed his first league start in just over a year.
Sears had come on as a substitute in games at Southampton, Sunderland and against Peterborough but was given the nod for a first start since Cotterill’s appointment - his first start since injuring his ankle against Newcastle under-21s in the EFL Trophy in September.
He replaced Donald Love at right wing-back, Love - who has impressed since his return to the fold - missed out through illness.
That was Town’s only change from the home victory last weekend. New signing David Davis made his first appearance in the squad since arriving on a free transfer from Birmingham and having subsequently completed a period of isolation.
Deadline day addition Curtis Main was signed too late to feature.
The Railwaymen made the short trip to Shropshire enjoying themselves in their first season back in the third tier.
David Artell’s men had won just once in seven games - drawing five - heading into the clash but it was a mark of how well they started the season that they were still 10th.
Their squad contained a number of familiar faces, as well as the boss. Centre-half pairing Omar Beckles and Olly Lancashire both had spells at Salop, as did former loan forward Mikael Mandron and substitute Chris Porter. Shrewsbury-born Billy Jones, on loan from Rotherham, was missing.
Shrewsbury - and Saturday’s match-winner Chapman in particular - picked up immediately where they left off against Posh.
From the very off Daniel Udoh won a good header and Shaun Whalley’s ball interested the lively Chapman, who nipped him in front of a defender and quickly found himself in on goal.
The Blackburn loanee left the Crewe backline in his wake. Into the box he was forced slightly wide by Alex goalkeeper Dave Richards and clever dinked finish dropped just the wrong side of the near post.
It was an electric start and Town continued in a similar vein. Town were visibly bubbly, sharp and confident both with and without the ball.
Sears made an assured start, reliable in possession and a real threat physically, as he won a couple of telling headers.
Josh Vela and Ollie Norburn continued their recent form in midfield, as they suffocated the play as Crewe attempted to play the ball out of their own six yard box and through their lines.
Nathanael Ogbeta looked to build on his very encouraging senior debut and was again assured early on with neat footwork and good delivery.
Town confidence looked high but, to their credit, the visitors grew into the contest. Daniel Powell shot low at vocal goalkeeper Matija Sarkic before Ryan Wintle shot well off target from distance.
Shrews got back on the front foot and Vela sent a decent header over after a good move involving Chapman, Udoh and Sears’ cross. Town were well on top as Vela sent a low shot at Richards seconds later.
But Crewe stuck to their guns of playing through lines and grew some confidence. Oli Finney’s shot deflected wide before the visitors led just ahead of the half hour mark.
The ball did not stick up front and right-back Travis Johnson delivered a cross that Ebanks-Landell, on the stretch, could only slice over him towards the back post, where Dale was in space to convert an easy finish.
Sarkic looked well-placed to collect the cross behind Ebanks-Landell but both communication and the clearance were lacking.
Centre-back Aaron Pierre almost sparked Town to an equaliser with a delightful run to the byline, his good low cross found Chapman, whose effort on the spin was blocked.
Whalley was typically industrious for the hosts and almost found himself in on goal but for Wintle’s defending.
A fine run to the byline from Vela, who dug out a brilliant cross that looped over keeper Edwards, was missing a telling touch at the back post just before the break as Town went into the interval behind.
Shrewsbury required some improvement and started the second period as they did the first.
Ogbeta found Whalley with a cool pass and Town’s No.7 arched a rocket from just outside of the Crewe box that flashed wide of Richards’ top left corner with the keeper scrambling.
Town stayed on the front foot and again caught Crewe trying to play short. Skipper Norburn shaped a well-struck 20-yard curler that was parried by a flying Richards.
Wilbraham and Longwell quickly rolled the dice, sending Josh Daniels on for Sears for a direct wing-back switch, but an attacking one.
And the hosts continued to threaten, with new-boy Ogbeta at the heart of a lot good from Town.
Pierre, Ogbeta and Pierre again were denied by solid Alex blocks before Ebanks-Landell headed over.
Ogbeta took things upon himself to haul his new side level. A barnstorming run into the left corner of the box was followed by a fierce rising strike that Richards tipped over. The left wing-back then shot wide from outside the box.
As Crewe took their time at every chance possible, Udoh nodded a Chapman corner wide at the far post.
Salop kept coming, Whalley’s low curled effort after a storming run down the left was parried by Richards and Crewe got lucky as the rebound dropped to a red shirt in the packed box.
Davis was introduced for his second Town debut with 12 minutes or so left, playing an advanced midfield role behind Whalley and Chapman.
But the final 10 minutes proved fruitless for Salop, who will file the evening under 'highly frustrating'.
Shrewsbury Town (3-5-2):
Sarkic; Williams (Pyke, 89), Ebanks-Landell, Pierre; Sears (Daniels, 57), Norburn ©, Vela, Chapman, Ogbeta; Whalley, Udoh (Davis, 78).
Subs not used: Burgoyne, Golbourne, Edwards, Goss.
Crewe Alexandra (4-3-3):
Richards; Johnson, Lancashire, Beckles, Pickering ©; Wintle, Finney (Lowery, 77), Murphy; Dale, Mandron, Powell (Walker, 77).
Subs not used: Jaaskelainen, Porter, Kirk, Adebisi, Evans.
Referee: Marc Edwards