The Bees bossed the first half and led 3-0 at the break. Rhodes’ superb opener set the tone after two minutes – an unstoppable volley from an acute angle high into the net.
Rhodes then slid in his second after Kelvin Langmead’s poor header, before adding a third from Charlie MacDonald’s pin-point cross.
Shrewsbury improved in the second half and pulled a goal back 20 minutes from time when top-scorer Grant Holt scrambled in his 21st goal of the season.
Town looked brighter in the closing stages, skipper Ben Davies coming close. And at the other end Shrewsbury goalkeeper Luke Daniels pulled off two smart saves.
The resolute Brentford defence and keeper Ben Hamer stood firm to see out the win as Andy Scott’s side kept up the pressure on leaders Wycombe. The defeat means Town slip out off the League Two play-off places.