Goals from Luciano Becchio, summer signings Luke Varney and David Norris and Ross McCormack's penalty secured an easy win.
Warnock named eight debutants plus teenage defender Sam Byram in a starting line-up which took the lead through Becchio in the 20th minute.
Another summer signing, Rodolph Austin, fired in a shot from outside the penalty area, and although Shrewsbury goalkeeper Chris Weale got down well to keep the ball out, Argentinian Becchio slid home the rebound from six yards.
Leeds doubled their lead six minutes later following a sweeping move that ended with striker McCormack's shot blocked and this time it was Varney who pounced to apply a neat finish.
Shrewsbury's Luke Summerfield lashed a shot wide for the visitors soon after, while team-mates Paul Parry and Jermaine Grandison had both gone close with long-distance efforts earlier.
But Leeds looked the more assured in possession and midfielder Norris went close to adding a third just before the break from 25 yards and again in the 56th minute from the edge of the box, before he slammed home a shot into the bottom corner from Varney's headed knockdown in the 66th minute.
McCormack stroked his penalty into the bottom corner to make it 4-0 in the 70th minute after Shrews defender Michael Hector handled in the area.