In awfully windy conditions at Sky Stadium, the Phoenix took it up to City and were only beaten by Wilkinson's 73rd-minute tap-in.
Matildas winger Holly McNamara gave Wellington's defence nightmares all game but left her shooting boots at home.
The Phoenix created several strong openings but were unable to capitalise.
Wellington teenager Manaia Elliott gave City some headaches down the left hand side in the first half, as did her replacement, Venezuelan forward Mariana Speckmaier, in the second.
City star McNamara hit a shot straight at Wellington goalkeeper Rylee Foster in the 21st minute and shortly after, Phoenix striker Emma Main forced a save from City's Lysianne Proulx.
Canadian Foster, who was playing her first game in 731 days, after suffering career-threatening injuries in a car crash, made a fingertip save in the 28th minute to prevent Wilkinson pouncing on an easy header.
Late in the half, McNamara dropped between the Phoenix lines and whipped a warning shot onto the roof of the net.
But Wellington kept pushing after the break and six minutes into the second half, Proulx made a superb save to deny Hailey Davidson.
In the 70th minute, Phoenix centre-back Kate Taylor broke away down the left on the counter and picked out Speckmaier, who blasted her first-time shot over the bar.
Three minutes later, City star Rhianna Pollicina forced a save from Foster but she could only parry it and a switched-on Wilkinson pounced to tap home.
City youngster Caitlin Karic spooned a close-range chance over the bar late in added time to make for a nervous finish.
But Wellington couldn't find an equaliser.
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