Goals from Amari Morgan-Smith and Robbie Willmott had put Luton into a 2-0 lead at half time, but it wasn’t until the final 10 minutes that the match really came to life.
Then, Freddie Murray, Jason Walker and Zdenek Kroca all netted, before Claude Gnakpa won and converted a penalty in the final moments.
Brabin, who had drawn his first three matches in charge, two due to last-minute equalisers, said: “It was a great way to get my first victory. It probably sounds easy now it’s happened, but you could see it all coming together, what we’ve tried to do.
“From a selfish point of view, I wanted my first win, first and foremost. The the players deserved it, everything I’ve asked of them they’ve responded brilliantly and theres a good atmosphere around the place.
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“We worked hard and we’ve changed our game a little bit and I thought it pieced together nicely.
“Full credit to Southport, they had a little period where I thought they put us under pressure and tested our defensive side that we’ve been working on.
“The lads did fantastically well and the atmosphere of the supporters, the fans were fantastic for us. Hopefully we keep going and keep building on that and peak at the right time.”
For more reaction and interviews with Zdenek Kroca, Freddie Murray and Alex Lawless, see this weeks Luton News/Dunstable Gazette, on sale Wednesday