The Magpies had the first opportunity of the game, with goalkeeper Callum James on hand to tip it behind.
Newcastle’s Joe White then smashed a volley at goal, testing James’ handling but it never phased the young goalkeeper.
Brown should’ve put Wanderers ahead 15 minutes into the game, with Argent-Barnes providing a tantalising ball into the area, but Brown nodded his header over the bar.
Argent-Barnes turned into a near deadlock breaker 25 minutes in, as he cut into the Newcastle penalty area before taking aim at goal, but had his effort blocked valiantly.
However, on the half hour mark, Argent-Barnes had his goal as he smashed the ball under Newcastle goalkeeper Daniel Langley’s body from an acute angle.
The young forward nearly made it two soon after, when he went through one-on-one with Langley and smashed the far post from 15 yards.
But come half time, Wanderers' lead was just one goal.
Following the break, Brown went close ten minutes in with a header on the Magpies’ goal forcing Langley to make a great diving save.
However, it wasn’t long before the striker doubled the Whites' lead.
Following a Wanderers attack where the ball rattled off the crossbar, Brown was first to follow up at the second chance and smashed it into the net.
Newcastle nearly provided the perfect response, as they charged down towards Wanderers' goal and smashed it goalwards, but James produced a delightful save to divert the ball onto the post.
The visitors clawed one back through Jack Young though, as his free kick found the back of the Wanderers goal.
With 12 minutes to go, Wanderers were making sure that their lead didn’t slip as James makes a quick reaction save before bodies flew in on the goal-line to send the ball away from danger.
With time ticking down however, Nicky Spooner’s men regained control of the tie and ultimately saw out the result until the final whistle.
Wanderers XI: James, Whalley, White, Broughton, Senior, Boon, King-Harmes, Connell (C), Brown, Darcy, Argent-Barnes
Substitutes: Graham, Alexander, Edmondson, Sterling, Richards