Shrewsbury were first to threaten when Kenny Gillet's weak headed clearance fell straight to Ben Davies, whose low shot was repelled by Rob Beckwith.
Barnet's stay of execution was, however, short-lived when more defensive indecision presented Guy Madjo with an invitation to shoot from distance.
The powerful forward's shot looped up off Ismail Yakubu's outstretched leg and over the stranded Beckwith to give the visitors a tenth minute lead.
But the lead lasted barely five minutes when Shrewsbury gifted Barnet a route back into the game.
Jason Puncheon's low cross wrong footed Kelvin Langmead, whose slip allowed Cliff Akurang to stab home his first goal for the club from six yards.
Akurang's confidence was clearly sky-high and, after a neat interchange with Albert Adomah, the towering striker launched a first-time effort which was splendidly tipped over by Glyn Garner.
Paul Fairclough's decision to revert to a more attacking 4-4-2 formation continued to reap dividends and it was no surprise when they took the lead in the 32nd minute.
Adomah linked up neatly in midfield with Joe Devera and the speedy winger ran unchallenged from halfway and fired a low shot across Garner and into the far corner.
Adomah repeated the trick in first-half injury-time when he latched onto Josh Wright's cute pass, stepped inside a couple of challenges and rifled his shot into the top corner.
Barnet started the second half in buoyant mood and went further ahead in the 54th minute when Akurang got his second, again from close-range after Adomah's cross had been touched back by Puncheon.