The visitors took the lead in fortuitous circumstances on 12 minutes when Manny Muscat headed Ki-Je Lee's corner into teammate Roly Bonevacia, and the midfielder was powerless to watch the ball roll into his own net.
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The Phoenix had an opportunity to get back into the game before the break when Roy Krishna was brought down just outside the box. Bonevacia took the set piece but he could only find the side-netting with his right-footed strike.
In the second half, Wellington did manage to get a goal back as Ben Sigmund raided down the right flank and cut the ball back into the middle. It evaded everyone bar Krishna, who turned it into the net.
With time running out in the game, the home side were given encouragement when the Jets were reduced to 10 men as Benjamin Kantarovski was sent off for his second bookable offence.
The Phoenix made Kantarovski pay in the dying seconds of the game when Nathan Burns was fouled outside the area, allowing Hicks to fire a free kick into the top corner past Ben Kennedy and earn his side a point.