A double from Andy Williams was preceded by goals from Michael Smith and Massimo Luongo as a three-goal salvo at the start of the second half earned all three points for the visitors.
Mark Cooper made two changes to the starting XI with Michael Smith and Rapha Branco returning to the side. Jon Obika and Yaser Kasim dropped to the bench with Jack Stephens moving into midfield.
It took Town’s Number 9 just three minutes to mark his return with a goal. A cross from the right wing found the striker inside the box and he swept the ball into the bottom corner of the goal.
With their tails up, Town continued to pour forward in search of a second and Nathan Byrne twice fired wide from distance.
As the hosts recovered from the early setback, they found a way back into the match on 18 minutes. Some neat passing in the midfield forced Jordan Turnbull into action to deny a certain goal-scoring opportunity. Unfortunately for him, the loose ball dropped at the feet of Bradshaw who steered it into the net beyond Foderingham.
Both sides saw efforts fly wide from distance before going in level at the break.
Barely a minute into the second half, Nathan Byrne picked off a Walsall corner and set off upfield on the break. With Massimo Luongo alongside him, the full-back squared the ball to his midfield counterpart and the Socceroos international drilled home.
Five minutes later, Town had opened up a two-goal cushion. Andy Williams played a one-two on the edge of the area. After racing onto the return pass, he cooly lifted the ball over the advancing goalkeeper and into the net.
Moments before the hour mark, the visitors made it four. Williams was again the beneficiary following a passing move which involved Byrne, Luongo and Michael Smith, tucking home his strike partner’s pass.
As the snowfall got heavier, the gameplay reflected the deterioration in conditions. However, Smith did have the ball in the back of the net once more but was flagged offside. Jack Stephens also smashed a free-kick wide.
At the other end, Wes Foderingham had to be alert to superbly tip away a free-kick that was bound for the corner of the net.