The unfamiliar Peter Johnstone Park added to the visitors' challenges.
"It didn't feel like a Stirling Sports Premiership venue but we have to work with whatever they can provide ... ," he said, adding it was fine once they had adapted as a small army of parochial fans added to the atmosphere on a day the Forsyth Barr Stadium in the city wasn't available.
The last-placed, winless hosts also provided striker Sam Mason-Smith with much-needed confidence through his two goals and, hopefully, something he would take to build on other venues, especially Park Island.
The opening goal came from Angus Kilkolly in the 17th minute, against the run of play, after the collective counterattacked from the spine of the field before exploiting the left flank to provide the striker with his second goal of the season.
Milne said Mason-Smith, who missed a chance in the 13th minute, was always going to find the net at some stage as someone who often created opportunities for others to score.
"It was only a matter of time so, yeah, it's honestly going to be good for his confidence ... so I'm really, really happy for him."
The Bay absorbed a 15-minute raid from Southern, who took advantage of the wind behind them and came close to scoring.
Goalkeeper Joshua Hill, who has become a cliche for the franchise team this summer, was the difference, mopping up anything the Milne-marshalled defence couldn't sweep up.
It was after Hill denied Southern's Andy Mulligan a goal in the 31st minute that an opportunity arose to tee up Bay United for a yawning space, again on the left flank for an enticing cross to Mason-Smith to finish in Kilkolly fashion for a 2-nil breather.
"We started a bit slow and they created a few chances but Josh was amazing again and made some great saves between moments," Milne said.
"That's what you expect from a great keeper and when you can do that it gives you the opportunity to punish teams, which we did so there was counterattacking play at times."
Understandably coach Brett Angell vented his spleen despite the 2-0 lead at halftime.
Milne said the lack of performance in the first half had ruffled Angell's feathers although he had reconciled that with their pole position to claim full points.
"I think he was more frustrated than upset that we didn't play as well as he would have liked for 45 but still managed to go up 2-nil into the wind as they tried to put us under pressure."
Bay United went into the second spell not necessarily with a bunker mentality but content to have the hosts toil harder into the wind trying to find goals.
Not surprisingly the third goal came in the 89th minute, again from a counterattack, with Mason Smith planting it into the net.
"We could have broken them down pretty quickly if we were on our game ... and dealt with those moments one way or another."
His troops also returned with a clean sheet on the officials' card without picking up a yellow card or any serious injuries, although bench striker Ben Everson didn't travel because of a back niggle.
The Bay are away to Hamilton Wanderers this Sunday in a televised match at the FMG Stadium, kicking off at 4.35pm.
The Wanderers copped a 4-nil flogging at the hands of Wellington Phoenix Under-20s on Saturday at home so they would be out to atone that.
"[We] dominated possession, created loads of chances and couldn't score," Southern coach Paul O'Reilly told Otago Daily Times.
"But again our Achilles heel seems to be that we can't put it together for a full 45 minutes or, certainly, a full 90 minutes. We switched off, two really poor goals from our point of view," O'Reilly lamented.
