Three Brentford goals in the first half-hour put the issue beyond doubt - as striker Jordan Rhodes helped himself to a hat-trick inside the first 30 minutes.
Although top scorer Grant Holt pulled one back after the break for Paul Simpson's men, their five-match unbeaten run came to an abrupt end as the visitors ran out comfortable winners.
After scoring in the first minute against Darlington last Saturday, Shrewsbury found themselves a goal down within 90 seconds as Rhodes opened the scoring with the pick of his three goals.
Barely a minute-and-a-half had been played when he crashed a superb dipping, angled drive past Shrews keeper Luke Daniels to give the high-flying Griffin Park outfit the perfect start - and when the Shrewsbury defence faltered twice during the first half he was on hand to make them pay.
After conceding just one goal in five matches since Boxing Day, the home side were breached again on 17 minutes when Rhodes pounced on a weak header from central defender Kelvin Langmead to slide the ball past Daniels from ten yards.
The Shrewsbury rearguard then failed to deal with an Adam Newton centre in the 29th minute, allowing Rhodes to nod home from eight yards and complete a quick-fire hat-trick.
For the first time in a League match this season Shrewsbury had conceded three goals and early in the second half there were more scares as Brentford remained in complete control.
However, the Shrews managed to rally and in the 68th minute Holt claimed his 21st goal of the campaign when he bundled the ball home from close range after Graham Coughlan had headed back across the goalmouth.
Despite some further pressure, Shrewsbury were unable to reduce the arrears any further as they slumped to their third home defeat of the season.