Wellington got the first shot away early on, coming from range as he hit a 25 yard rocket that went right at Eugene Galekovic.
Both sides struggled in the early going, with neither side able to take control, a situation especially frustrating for the visitors who thrive on holding possession.
With Adelaide not getting in to their usual rhythm, it allowed Wellington some early chances, and in the 19th minute Cunningham attempted to chip Galekovic, only for his effort to go over the bar.
Adelaide were trying to use their wings, but just couldn’t connect on that final pass, which let them down far too much during the night.
Stein Huysegems could well have scored the opener in stoppage time but he was denied after being played through as his touch was much too tame for Galekovic.
Adelaide looked much better in the second half as they started to click, with Michael Zullo and Bruce Djite looking much more active for the away side.
Zullo and Tarek Elrich combined well on the wing but it seemed Elrich was holding on to the ball for too long, only for Andrew Durante to slide in and foul Elrich, leading to a penalty.
That allowed Marcelo Carrusca to step up and instead of powering the ball, he chipped it right up the middle, producing a Panenka that left Glen Moss looking foolish and giving Adelaide the vital goal.
Jeremy Brockie nearly found an equalizer almost instantly as his effort from distance was well saved by Eugene Galekovic to keep the Reds in front.
Zullo should have wrapped things up in the 75th minute as he was sent free, but instead of shooting he tried to find Carrusca, failing to do so, allowing Wellington to clear.
Wellington tried to come back in the latter stages, but the Adelaide defense was much too solid, keeping the hosts out and holding on for a vital 1-0 victory.