Goals from Tom Craddock, Sam Parkin and Asa Hall were enough to give the Hatters a precious three points while the Shrews, who struck through Ben Davies, were left without an away league win since August.
Shrewsbury started the brighter with Grant Holt flashing a 16-yard shot wide in the third minute, but it was the hosts who went ahead in the 15th minute from their first corner.
Goalkeeper Luke Daniels was forced to palm behind Claude Gnakpa's curling cross and, from Rossi Jarvis' resulting flag kick, Hall's near-post flick-on allowed Craddock to tap home his sixth of the season from a yard out.
The Shrews were behind for just four minutes as Terry Dunfield's chipped pass was chested down by Richard Walker into the path of Davies, who strode into the penalty area and prodded home into the far corner with his right foot for his 11th of the campaign.
The Hatters then forced a period of pressure with Daniels tipping over Hall's 30-yarder at full stretch in the 22nd minute before Gnakpa headed Jarvis' corner just wide seconds later.
Daniels had to be quick off his line to head clear from Parkin in the 27th minute and, from the follow-up, Craddock could only put his 25-yard effort wide.
Shrewsbury came back and Chris Humphrey's ferocious 22-yard half-volley came back off the crossbar in the 29th minute after Davies' free-kick from the left had only been half-cleared.
Luton went back in front in the 35th minute when Sol Davis' tackle set up Chris Martin to cross into the box for Parkin to control the ball on his chest before lashing home from ten yards with his left foot for his first goal since August.
Holt should have levelled a minute before the break but, after playing a neat one-two with strike partner Walker, he scuffed his left-foot shot wide from eight yards.
Five minutes after the interval Holt was again played clean through this time by Dunfield's delicate lob, but the former Nottingham Forest man was denied by a fine point-blank save from Hatters stopper Dean Brill.
Craddock sneaked an opportunity for Luton in the 59th minute after he robbed Neil Ashton and then bobbled an 18-yard effort just past the far post.
Shrewsbury again came close to a leveller on the hour mark as Ed Asafu-Adjaye was pulled up for handball and Davies curled the 30-yard free-kick off the underside of the crossbar.
Holt spurned yet another opportunity for the visitors in the 70th minute as he was left all alone to run on to Ashton's hopeful long punt only to touch the ball past Brill but also past the post.
Holt's misses were made all the more painful for the Shrews three minutes later as Luton bagged a third when Michael Jackson's defensive header fell for Hall to hit first time from 22 yards with the ball nestling in the net off the foot of the post.