Having dominated the game from around the half-hour mark, the Phoenix were unable to find a goal to reward their ascendancy and looked as though they'd rue that profligacy when Mileusnic controlled a ball on his chest on the corner of the penalty area and without letting it bounce, fashioned a splendid, dipping left-footed volley into the top corner to put his side ahead.
But Wellington found an equaliser ten minutes later when Ball tapped home Josh Sotirio's low cross, with Adelaide goalkeeper Paul Izzo probably feeling he could have done better.
Shortly afterwards, the Phoenix were awarded a penalty when Liberato Cacace went to ground in the area. The VAR checked and upheld the decision of the referee, with Davila making no mistake from 12 yards, netting his fifth goal of an increasingly impressive season.
Phoenix coach Ufuk Talay made just one change to the starting side that beat Brisbane a week ago, replacing Callum McCowatt with Australian attacker Reno Piscopo, who almost had a dream start to the game, seeing his close-range header saved by Izzo after just 30 seconds.
From there, Adelaide went on the front foot, dominating possession and asking plenty of questions of Wellington's defence through a succession of corners and attacking moves.
But the Phoenix held firm, grew into the game and were awarded a penalty after half an hour when Sotirio was brought down by Michael Jacobsen. On closer examination by the VAR, the foul was adjudged to have occurred marginally outside the penalty area.
While that chance was lost, it was the catalyst for an excellent performance from the visitors, with their crisp passing and flowing football as good as anything they've produced this season.
Alex Rufer was a gigantic presence in Wellington's midfield, winning tackles and releasing his teammates with accurate short passes, while Piscopo, Davila and the tireless Ball were constantly involved in the front third.
The home side were heavy favourites, having won their last four games, while Wellington had won on just three of their previous 20 visits to Adelaide, the most recent of which was on New Year's Eve 2014.
However, with 26 shots to eight, the Phoenix totally deserved this win and next host the Western Sydney Wanderers at Eden Park on Saturday night.
Wellington Phoenix 2 (Ball 67', Davila pen 71')
Adelaide United 1 (Mileusnic 57')
