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After a goalless 75 minutes, the home side always looked liked making the breakthrough but struggled to convert any of the chances from inside the Adelaide box.
However, the game started to open up with 15 minutes left and it was substitute Griffiths who broke the deadlock, after some neat wing play from Louis Fenton found the forward inside the six-yard box for him to put it past Eugene Galekovic in the United net.
Burns then doubled Wellington's lead two minutes later and wrapped up all three points, after a smart set piece found the striker all alone in the box with only the keeper to beat.
The result sees Phoenix temporarily go top, whereas Adelaide fall further down the table to fifth, despite being only three points off the leaders.