Neil 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 after a defensive mix-up.
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 after another mistake by the Shrews defence.
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 from a Paul Green effort.