The Bulls fell behind after half an hour, but showed plenty of resilience to battle back to record their first league success against their old adversaries for almost 32 years.
Hereford controlled the early stages and Sam Gwynne saw the ball taken off his toe just as he was about to lash home Simon Johnson's cross.
Shrewsbury were rarely able to penetrate the home defence but took the lead against the run of play on 31 minutes.
Keith Briggs received the ball on the right of the Hereford area and marked his debut by firing in a shot which deflected off Dean Beckwith past Wayne Brown.
Trevor Benjamin appeared to be brought down in the area by Kelvin Langmead, but no penalty was given and, as Hereford continued to press, skipper Ben Smith shot against the post.
But, four minutes before the break, an innocuous challenge by Marc Tierney on Theo Robinson brought a penalty, which Benjamin smashed past Glyn Garner.
The visitors played some controlled football into the wind in the second period and almost regained the lead on 67 minutes.
Kevin McIntyre's corner was headed by Langmead towards the corner of the net, but Gwynne leapt high to nod the ball off the line.
Hereford made their Shropshire rivals pay from a corner of their own at the other end, when Beckwith directed Kris Taylor's flag-kick into the net via a defender.
United then began to find holes in the Town defence and extended their lead on 79 minutes. Smith slipped a pass through to Johnson, who slid his shot into the corner of the net from the edge of the area.
The winger almost added a fourth shortly afterwards when he raced on to another Smith pass, but this time directed his effort just wide.