Paul Warne’s Millers were in control almost throughout as they convincingly secured their passage into round two.
The Millers absolutely flew out of the traps as they were two goals up after only three minutes.
Kyle Vassell touched the ball to Matt Crooks and he opened the scoring from close range.
Barely a minute later the lead was doubled when Vassell lashed home a clinical shot past ‘keeper Joe Murphy.
The visitors kept their foot on the gas and almost went further clear in the 11th minute when Michael Ihiekwe glanced a header narrowly wide.
The Shrews finally caught sight of goal when Brad Walker’s powerful strike was deflected wide by Richard Wood.
Shortly before the break it was 3-0 when Freddie Ladapo darted onto substitute Michael Smith’s header to fire home comfortably. Ladapo almost made it four shortly after the restart as his powerful drive stung the palms of Murphy. The Shrews’ ‘keeper then stuck out a leg to deny Smith a fourth. The Millers made it four when Wood smashed home from 12 yards.
Shrewsbury: Murphy, Barnett, Beckles, Ebanks-Landell, Waterfall (Pierre, 5), Walker, Rowland, McCormick, Golbourne (Whalley, 46), Morison (Giles, 78), Okenabirhie. Unused substitutes: O’Leary, Love, Taylor, Vincelot.
Rotherham United: Price, Olosunde, Wood, Robertson, Ihiekwe, Wiles (Lindsay, 76), Barlaser, Crooks, Morris (Smith, 6), Ladapo (MacDonald, 64), Vassell. Unused substitutes: Iversen, Mattock, Southern-Cooper, Proctor.
Referee: A Backhouse.