Cotterill’s side, which featured 10 changes from Saturday’s league draw, were second-best throughout and the scoreline reflected the Imps’ dominance in the round of 32 tie.
The tie, a repeat of the 2018 EFL Trophy final, was a dress-rehearsal for next Tuesday’s league match, which is a clear priority for the League One strugglers as Cotterill suffered his first defeat in charge after four games.
Goals from Anthony Scully, Zack Elbouzedi, Remy Howarth and a Jorge Grant penalty did the damage. Shilow Tracey netted on the stroke of half-time to bring Town back in it at 2-1.
But Shrews were soundly beaten on an evening where Cotterill ran the rule over his supporting cast.
Cotterill made 10 changes to the Shrewsbury side that earned a crucial last-gasp League One point against Charlton. Ethan Ebanks-Landell was the only survivor of the previous clash.
Headline inclusions to the home side’s XI were a first start under Cotterill for Jason Cummings, while Tracey - who bagged a hat-trick in the competition last time out - returned from injury.
Donald Love, who is not registered for league action as part of Salop’s 22-man squad, was offered his first run-out since the previous Trophy tie. Harry Burgoyne returned in goal for the rested Matija Sarkic.
Scott High, Jan Zamburek and Josh Daniels were also given chances to impress as Cotterill rang the changes.
Ryan Sears’ place on the substitutes bench came as a boost to those of a Town persuasion, the defender has been injured for three months since injuring his ankle against Newcastle under-21s in the Trophy.
Michael Appleton’s Lincoln, second in the third tier, made six changes from their 2-0 win at Rochdale on Saturday. They named just five substitutes.
But, as would prove the theme, the Imps quickly took charge.
The boss was unhappy as High was turned easily in midfield and Scully’s delivery from the right was met by Elbouzedi’s back-post header. The attacker did everything right, heading back the way it came, but Burgoyne’s brilliant outstretched left hand made the save.
The Imps were well on top. Scully clipped the angle of post and bar with a fierce 25-yard free-kick before Elbouzedi and Robbie Gotts both at Burgoyne.
The lead Lincoln threatened came on 18 minutes. Elbouzedi sped half the length of the pitch clear of Daniels and Love and slipping in Scully who finished tidily.
Shrews were unconvincing but a neat move down the left involving Zamburek ended in High drawing a neat save from Alex Palmer with an effort on the spin.
Lincoln were the more settled and fluid, playing with visible confidence. Howarth should’ve done better in the box before the lively Scunny bent a stunning curler that drew a top-drawer fingertip save from Burgoyne.
Cummings shot over from an angle after lively play from Tracey but it was against the run of play as the visitors doubled their lead five minutes before the break.
Brenan broke powerfully from midfield before spreading to Harry Anderson on the right. The winger’s delivery to the back post was nodded in by Elbouzedi, who had escaped Love.
It looked like a long night ahead for the hosts but Shrews knitted together a neat move on the cusp of half-time to pull one back.
Daniels centred for Cummings whose neat lay-off to Tracey was swept confidently into the bottom right corner.
Cotterill sent Ro-Shaun Williams on for Ebanks-Landell at the break. Some more haphazard defending allowed Howarth an early sighter but Burgoyne saved.
Town enjoyed a good spell inside the first 15 minutes of the second period. They probed with an extended spell of possession but could not force an opening of note.
But Lincoln on-loan Forest midfield man Johnson broke brilliantly from midfield and Howard was afforded far too much room to slot in a simple finish from 10 yards as the Imps put the tie to bed midway through the half.
A couple of hundred Town fans, dotted in the West Stand, attempted to inject some urgency as Dave Edwards headed Love’s cross over.
But Williams clipped the impressive Johnson in the box and half-time sub Grant dispatched the spot-kick to put a gloss on the scoreline on an evening to forget.
Shrewsbury Town (3-5-2):
Burgoyne; Love, Ebanks-Landell (Williams, 45), Golbourne; J Daniels, Edwards ©, Goss, Zamburek; High (Barnett, 65); Tracey, Cummings (Caton, 85).
Subs not used: Iliev (gk), Sears, Lloyd, Aris.
Lincoln City (4-3-3):
Palmer; Gotts, Eyoma (Cann, 83), Melbourne, Roughan; Haworth, McGrandles (Grant, 45), Johnson; Anderson, Elbouzedi, Scully.
Subs not used: Long (gk), Hopper, Walsh.
Referee: Ben Toner