Following a hopeful through-ball, the Shrews left-back got himself into a complete mix-up in defence and in trying to play the ball back to Luke Daniels, inadvertently lobbed his goalkeeper from over 20 yards.
In truth, the result flattered Bury - who failed to have a meaningful effort on target throughout the entire encounter.
Town did not muster much more in an open game devoid of chances and a clear lack of quality in the final third - despite the presence of Grant Holt and Andy Bishop, who have scored 45 goals between them this season.
Ben Davies forced a good stop from Wayne Brown in the first half before Holt saw his follow-up effort hit a post, the latter also being denied by Brown late on.
The odds are now against Shrewsbury, who managed just three away wins all season, progressing to Wembley, but the manner in which they clinched their play-off place at Dagenham on the final day will provide hope.
Bury, on the other hand, bounced straight back from their heart-breaking end to the season and will fancy their chances in the second leg at Gigg Lane on Sunday.
Shrews seized the early initiative but neither goalkeeper was seriously threatened inside the opening 10 minutes.
Bury soon got a foothold in the game, though, and despite Brian Barry-Murphy failing to connect with a low pass from 10 yards out, were looking the more threatening as the first half wore on.
Yet the 8,429 inside the Prostar Stadium, the club's biggest attendance of the season, had to wait until the 33rd minute for the first real chance of the encounter.
Town winger Chris Humphrey burst down the right and squared for 13-goal midfielder Davies, but he could only watch as Brown parried his powerful shot to safety.
But the ball was immediately recycled to Holt, who fired a second effort off a post as the first half ended goalless.
The second half got under way with a lot more urgency but as was the case in the first half, neither side could fashion a clear-cut opening.
Michael Jones whistled a free-kick narrowly wide of the upright in the 67th minute, but Shrews keeper Daniels had still not been called upon to make a single save as the game approached the final 20 minutes.
Striker Omer Riza was introduced for the home side as Simpson went in search of a crucial first leg advantage.
Controversy reigned as the game entered the closing stages as Bury captain Ben Futcher appeared to haul back Riza, with the substitute pressing to go clear through.
Then, with nine minutes remaining, Ashton's howler handed the visitors the crucial advantage.
Holt was then lucky to stay on the pitch as he grabbed Steve Haslam around the throat as tempers boiled over.