Tyler Boyd was Wellington's hero, with the 20-year-old scoring a late brace after coming on with just 19 minutes remaining to earn his side a valuable three points in the division's final match of 2014.
Kenny Cunningham gave the visitors the lead in the 54th minute with his first A-League goal of the season.
However, Adelaide looked on course to take a point when Spanish forward Cirio equalised 20 minutes later.
But Boyd - who came on for Cunningham - had other ideas.
The youngster restored Wellington's advantage in the 77th minute, before sealing the victory with his second in stoppage time.
In-form Wellington are now unbeaten in six A-League outings, and sit just four points adrift of table-topping Perth Glory.
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