Graham Turner's men, stung by losing their proud 34-match unbeaten home run at the weekend, came flying out of the blocks and took the lead as early as the third minute at the New Meadow.
Paul Parry floated in a free-kick and big defender Darren Jones looped a header over Sky Blues goalkeeper Joe Murphy.
Shrews controlled the first half for long stages and it was no surprise when they doubled their advantage in the 21st minute. Parry took aim with a 25-yard free-kick which took a slight deflection off team-mate Matt Richards on its way to the roof of the net, although Parry was credited with the goal.
Richards increased the lead in the 62nd minute, sending Murphy the wrong way from the penalty spot after Luke Summerfield's surging run into the box was crudely ended by William Edjenguele.
It was quickly four-nil to rampant Shrewsbury as Marvin Morgan powered past defender Richard Wood to hammer home. Only then did Coventry come to life although a 75th-minute penalty from John Fleck, awarded after Jermaine Grandison upended substitute Gary McSheffrey, was little more than consolation.