Barrow started the brighter of the two teams, but they could do nothing about Drury's left-foot stunner from 20 yards which flew beyond keeper Stuart Tomlinson.
The goal hit the Bluebirds hard, and they could have gone further behind but a goalmouth scramble ended with the ball being hacked clear by a frantic Barrow defence.
But Bridges' sending off was a turning point in a feisty clash, and it allowed Barrow to gain the possession they were otherwise struggling to find.
Simon Wiles and Paul Rutherford began to exploit the space in the final third as they looked to carve Stevenage open, although Boro keeper Chris Day remained largely untroubled.
The blistering pace of Yemi Odubade provided a constant outlet for Boro, the former Oxford United man being unfortunate not to win a penalty after a hefty challenge from Paul Edwards.
The Hertfordshire side almost won it when Jon Ashton flashed a header wide from a corner.
The miss proved crucial when, moments later, substitute McEvilly, who had only been on the pitch for five minutes, planted his head on a Rutherford cross, guiding the ball beyond a hapless Day.
The game suffered its second sending off deep into stoppage time when Hulbert's elbow led to Griffin being taken off on a stretcher. And, with Stevenage having used all their substitutes, Graham Westley's side had to soldier on with nine men.
With the additional space created by the two red cards and the Griffin injury, both sides looked to exploit the width of the pitch in extra time and first Odubade and then Edwards came close with rasping shots that flashed just wide.
But it was left to local lad Walker to fire his side to FA Trophy glory with that blistering effort that sent the Cumbrian fans into raptures and crown his season as Barrow's 21-goal top scorer.
Barrow boss Dave Bayliss told BBC Radio Cumbria:
"To go a goal down in the way we did, when we were the better side, was hard. but we told the lads to keep going.
"Stevenage are a top side, but we just kept going and got our reward in the end. The players have worked extremely hard this season and they deserve it."
Stevenage manager Graham Westley told BBC Three Counties Radio:
"Circumstances conspired against us today. I thought we were valiant in our effort. The players worked their socks off for each other.
"I thought in the end the legs that we didn't have out on the pitch because of the Griffin injury probably cost us.
"Credit to Barrow, they have won the Trophy and they deserve to enjoy themselves."
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