The 30-year-old tapped home Jordan Shipley’s cross just four minutes after coming off the bench to cap a clinical performance from Steve Cotterill’s team.
Christian Saydee had earlier put Town ahead against his former club on his full home debut.
Burton, who dominated for long spells, looked favourites to go on and win when Victor Adeboyejo equalised just before the hour mark, only for Bowman to steal the show.
The fact Shrewsbury’s goals, in the 13th and 66th minutes, came with their first two efforts on target says everything about a match where they were for large parts second best.
Goalkeeper Marko Marosi was undoubtedly the star man, preserving the win with a brilliant save to deny Brewers sub Sam Winnall.
Yet in a week where Cotterill called out his strikers, there was some irony in the clincity of his team’s display.
The manager should also take credit for a tactical switch when the score was 1-1, from a back four to a back three, which helped deliver not only the win but a much more cohesive performance for the final half-hour.
It was Burton who made the brighter start, Marosi forced into action inside three minutes tipping Sam Hughes header from Tom Hamer’s cross over the bar.
But when Saydee got his first sight of goal, he did not hesitate. Tom Bayliss flicked on Matt Pennington’s long ball into the 20-year-old's path and after taking a touch to steady himself he drove a powerful finish beyond Ben Garratt and into the bottom corner.
Shrewsbury were ahead but a long way from comfortable. Davis Keillor-Dunn should have done better than shoot weakly at Marosi after a misdirected pass from Jordan Shipley, drafted into the starting XI in place of the unavailable Tom Flanagan, wrong-footed the home defence. Pennington was then in the right place at the right time to clear when a cross from the right threatened to bounce through to the waiting Victor Adeboyejo.
Just past the half-hour mark, Hughes thought he had levelled when he rose to meet Hamer’s long throw at the far post, only for George Nurse to appear from nowhere at the far post and clear off the line. There was further drama in the home box when the ball fell for John Brayford 10 yards out but his well-struck volley thudded against a defender and to safety.
Just as in the first half, Marosi was forced into early action in the second. This time his stop from Jonny Smith’s low drive sent the ball thudding into Hamer and wide of the target.
Marosi’s next save was the best of the lot. Deji Oshijala looked certain to score arriving at the back post but the keeper flung himself low to his right and turned the ball round the post.
The keeper collided with the upright making the stop and there was a delay of around five minutes as he received treatment.
Yet the restart brought no respite for the hosts and finally Burton’s pressure paid off. Oshijala got free again on the left and though Marosi was again equal to the shot, this time the rebound struck Adeboyejo and flew into the net.
Adeboyejo then came close to putting the visitors ahead when he shot wide from the edge of the box.
But it was Town who instead got their noses back in front, with virtually their first attack of the second half in the 66th minute.
Tom Bayliss picked out Shipley’s run with a well-weighted through ball and the former Coventry man, now playing as a left wing-back, crossed for the recently introduced Bowman to tap home.
Town were far better thanks to the tactical switch to three at the back but it still required more brilliance from Marosi to deny Brewers sub Sam Winnall at point-blank range with five minutes remaining.
The hosts were a long way from home and hosed, with referee Thomas Kirk somehow calculating 13 minutes added time.
Oshijala smashed the ball across the face of goal five minutes into the extra period, with Adeboyejo somehow failing to get the vital touch. The latter then hooked over following a scramble as the hosts hung on.
13 GOAL Town take the lead. Christian Saydee latches on to Tom Bayliss’ flick and finishes powerfully into the bottom corner.
58 GOAL Burton pull level. Deji Oshijala’s shot is saved by Marko Marosi but the ball hits Victor Adeboyejo and flies in.
66 GOAL Town back ahead. Bayliss picks out Jordan Shipley and he crosses for substitute Ryan Bowman to tap home.
Town (4-3-3): Marosi, Pennington, Dunkley, Moore, Nurse (Bowman 62), Winchester, Leahy, Bayliss, Pyke (Bennett 62), Saydee (Street 76), Shipley Subs not used: Bloxham, Caton, Craig, Burgoyne (gk).
Burton: Garratt, Hamer, Hughes, Brayford, Borthwick-Jackson, Smith (Winnall 82), Taylor (Onyango 90), Oshilaja, Powell, Adeboyejo, Keillor-Dunn (Carayol 73) Subs not used: Thorpe, Lakin, Hewlett, Kean (gk).