Shakers striker Newby replaced John Fitzgerald on the hour mark, and it took the former Huddersfield hitman just nine minutes to make a lasting impression, playing his first game in five months because of an ankle-injury.
Newby headed home after Allan Smart had crossed from the left and midfielder Dwayne Mattis had headed the ball into his path.
Newby was greeted with plenty of applause from the Bury fans on his late arrival - and that turned to rapturous applause for rescuing what had been, up to then, a frustrating contest for both sides.
Bury sealed the points in the final minute when Shrewsbury goalkeeper Joe Hart made a mess of a clearance under pressure from Simon Whaley, and when the ball fell at the feet of Barry-Murphy he hit the target with a wonderful left-footed 40-yard strike.
Mattis did well to direct his header back into the six-yard box and although Shrewsbury defender Dave Walton nearly cleared off the line, Newby's header was adjudged to have gone over by the linesman on the far touchline.
Having lost both of their opening fixtures 2-1, Barrow was pleased that his side finally managed to get a win under their belts.
Bury started the better of the two sides and should have taken the lead in the 12th minute, but Barry-Murphy saw his diving header rebound to safety off the crossbar.
The home side continued to press and came close to breaking the deadlock eight minutes later through striker Smart.
However, Smart shot wide from a tight angle and then soon after Whaley, playing in attack instead of his preferred midfielder role, supplied his fellow striker with a good ball, but again a chance for an opener went begging.
Dave Flitcroft headed over in the 25th minute and seven minutes before half-time former Stockport County full-back John Hardiker saw a shot saved by Hart.
Shrewsbury had their first chance in the 39th minute, but Bury goalkeeper Neil Edwards saved well from a left-footed drive from Ben Smith.
Barry-Murphy nearly gave the Shakers the lead three minutes before the interval, but his clever swivel-shot was well-saved by Hart when it looked for all the world as if the ball was going to hit the top left-hand corner of the net.
Whaley also went close just before half-time, his curling shot flying inches over the crossbar having avoided the attentions of two Shrewsbury defenders.
The second-half was quiet in comparison to the first, at least for the number of chances.
Whaley shot over in the 49th minute and Smart saw a long-range drive easily saved by Hart in the 65th minute as Barrow's men continued to push for a breakthrough.
At the other end Colin McMenanin crossed for Mark Stallard, but the Shrewsbury striker just failed to get the slightest of touches which would have surely seen the ball in the back of the net.
Shrewsbury so nearly grabbed a share of the spoils with six minutes remaining, but Neil Sorvel could only look on with dismay as his 25-yard volley thundered into the crossbar and rebounded to safety.