Jack Muldoon broke Barrow hearts for the second time this season as his 84th minute goal made it three Football League wins in a row for Town.
Despite attacking intent being shown by both, the fixture looked destined for a goalless draw before substitutes Muldoon and Josh Falkingham combined on the counter attack to seal a late win and keep Town's recent run of 1-0 victories going.
Weaver made two changes from Saturday's win over Grimsby as Mark Beck and Warren Burrell replaced Aaron Martin and Connor Hall, the latter sidelined with a hamstring injury that is expected to sideline the centre half for around ten days.
Scott Quigley had the first chance of the evening and should have found the back of the net after getting on the end of a low cross from the right, though the offside flag spared the forward's blushes.
Town went close next as Josh March closed down Joel Dixon and managed to get something on the ball, only for it to bounce wide of the target.
The sides continued to exchange chances with Quigley forcing Belshaw into a save with his feet at his near post, moments before Josh McPake cut inside with purpose and saw his strike beaten away by Joel Dixon.
March felt he should have earned his side a penalty five minutes before the break as he got the better of James Jones and looked to have been brought to the ground, but the Referee waved away the protests as the sides went into thr interval level.
The second half started with the same attacking intent both sides showed in the first 45 as Quigley again produced a good save from Belshaw, before a loose Dixon pass presented an opportunity for March, but the Town attacker could only shoot straight at the Barrow 'keeper.
With Jack Muldoon preparing to replace him, March's last involvment almost gave his side the lead as he sent a low shot goalwards that was saved by Dixon. Martin and Josh Falkingham were also introduced for the yellow carded Lloyd Kerry and Beck.
It was the hosts who would go clode next though with Quigley's touch letting him down as he ran through on goal before Jamie Devitt tested Belshaw with a long range effort.
With six minutes of normal time remaining, Jack Muldoon popped up with what would turn out to be the winning goal, latching onto Falkingham's through ball on the counter attack and calmly finding the corner.
Town: Belshaw, Fallowfield, Jones, Smith, Burrell, Thomson, Beck (Martin 63), Kerry (Falkingham 63), March (Muldoon 63), McPake, Power
Subs not used: Lawlor, Francis, Minter
Goals: Muldoon 84
Booked: Kerry, Smith
Barrow: Dixon, Brough, Taylor, Quigley, Jones, Nthle (Devitt 60), Barry (Ndjoli 91), Eardley, Thomas, James, Banks (Taylor 59)
Subs not used: Lillis, Goodridge, Donohue
Referee: Robert Lewis