The Sulphurites chalked up their fifth consecutive victory on the road on Saturday, a result which lifts them to 14th in the League Two standings where they now sit just five points outside the play-off places.
Having started slowly before growing into the first half, Town again found themselves on the back foot midway through the second period, which was when boss Weaver decided to utilise the strength he had held in reserve.
The introduction of Dean Cornelius, Liam Gibson and George Thomson helped freshen things up for the away side, with Thomson going on to decide the contest with an 82nd-minute strike from distance.
"First half, I felt a bit anxious, I could feel the anxiety in the players because we weren't in control of it, we were giving too many needless free-kicks away, there were too many entries into our box," Weaver reflected.
"But, we defended well, we looked a hard nut to crack and, with that, we grew in confidence and always looked a threat on the break.
"The lads who came off the bench made a telling impact on our situation because their crowd were just starting to get behind them. They were a bit on top, getting balls into our box and we weren't quite in control.
"So, looking at the quality and experience on our bench it just felt time to make substitutions, slow that momentum, and I'm putting on good players.
"I was pleased to be able to bring such strong players on, and with them fresh, we were able to create a momentum-changing moment in the game and struck gold with Thommo."
Town’s winner arrived after right-back Kayne Ramsay played the ball inside and Thomson drove towards goal before unleashing a left-footed effort from 25 yards out, beating Walsall stopper Owen Evans at his near post.
"It wasn't his finest strike, but it went in and as we keep saying, it's about getting your shots off,” Weaver said of the game’s decisive moment.