Karleigh Osborne's headed goal early in the second half gave the West Londoners a victory they thoroughly deserved as they recorded their third straight win - while sentencing a lack-lustre home side to a third consecutive defeat as they continue to lose ground in the play-off stakes.
Following successive defeats against Wycombe last Saturday and at Peterborough on New Year's Day, this has been a week of upheaval for Shrewsbury as influential midfielder Stewart Drummond - scorer of their goal in the League Two play-off final last term - departed for Morecambe, while joining Town from Macclesfield was midfield man Kevin McIntyre, who was handed an immediate debut.
Following the departure of Drummond, winger Ben Davies took over the captaincy.
There was little to concern either goalkeeper during a low-key first half in which Brentford had the greater share of possession without mounting much in the way of a serious goal threat.
Shrewsbury's forward play was even worse and it took until first-half stoppage-time for the them to manage an effort on goal, but a long-range Davies effort flashed wide of the post.
After the break the visitors continued to hold the upper-hand and deservedly went ahead seven minutes after the restart. Ryan Dickson floated in a corner from the right and central defender Osborne rose unchallenged to nod home from seven yards.
On the hour Bees skipper Matt Heywood cleared an angled Davies drive off the line and, just before being substituted, veteran striker Andy Cooke pulled a shot wide.
However that was all Shrewsbury could muster in response and ten minutes from time Brentford missed a chance to increase their advantage when top scorer Glenn Poole fired over from close range.