The home side had the better of the chances, forcing returning stopper Joe Hart into two big saves in the first half of an otherwise quiet contest.
Three-time winners West Ham barely got out of first gear all afternoon, rarely registering attacking attempts to add another fixture to their hectic schedule.
The opening quarter of the match passed by with little incident, with both sides managing to cancel each other out in what was proving to be a tight cup tie from the off.
League One promotion chasers Shrewsbury grew more and more into the contest, however, and had a couple of big chances to open the scoring.
Matt Sadler found himself in an advanced position and had only Hart to beat from the angle, but the former Shrewsbury stopper was equal to it down low.
A couple of minutes later Hart, who appeared 58 times for Salop as a teenager, was equal to an Alex Rodman effort after the wideman cut in from the left-hand side.
Town created the first half's better chances, seeing a dangerous back-post cross pounced on by Hart and Omar Beckles blasted over the bar from the edge of the box.
A tired West Ham side, showing just three changes from last time out, struggled with the pace of the game, meanwhile, with their only attempt of any note a Declan Rice cross that Cheikhou Kouyate could not divert on target.
Visiting boss David Moyes did not have any real outfield experience on his bench to call upon at the break, with just six senior appearances between the subs, and the Scotsman stuck with what he had until the 70-minute mark.
Chances were at a premium up until that point as, a glanced Andre Ayew header aside, neither keeper had anything of note to do.
Shrewsbury, boasting the joint-best defensive home record in English football's top four leagues, continued to stifle their opponents as the contest entered its final quarter.
The only clear-cut chance of the second half arrived at the feet of Jon Nolan five minutes from time, but he could only blast a cutback over the crossbar from 10 yards out, meaning that the sides will face off again in a little over a week's time.
SHREWSBURY TOWN (4-5-1): Henderson; Beckles, Sadler, Nsiala, Bolton; Rodman (Lowe 76'), Ogogo, Godfrey, Nolan, Whalley (Gnahoua 88'); Morris (Payne 82')
WEST HAM UNITED (5-3-2): Hart; Burke, Rice, Reid (Quina 86'), Ogbonna, Masuaku; Kouyate, Obiang, Cullen; Ayew, Hernandez (Martinez 71')