Stein Huysegems scored the only goal of the game after 33 minutes, following a superb pass from Carlos Hernandez.
Huysegems also hit the post as Wellington saw off a Sydney side sorely missing injured Italian star Alessandro Del Piero.
The hosts saw Paul Ifill leave the field on a stretcher with an Achilles injury early in the second half, but held on to give Ernie Merrick a maiden success as coach.
Sydney entered the game on the back of five wins from six matches, but missed the opportunity to go second in the table.
In Sunday's other game, Ryo Nagai suffered a hamstring injury as Perth Glory played out an uninspiring 0-0 draw with Adelaide United at nib Stadium.
Nagai was withdrawn in the 72nd minute as both sides - seventh and eighth in the table respectively - searched in vain for a late winner.
The Japanese winger joins a long injury list for Perth, who sacked coach Alistair Edwards this week amid a bitter feud with captain Jacob Burns and other senior players.
Burns, who confronted Edwards after last week's 2-0 loss to Melbourne Victory over perceived nepotism towards Edwards' sons Ryan and Cameron, was booed by a small section of the crowd at the start of the match.
But those jeers became cheers in the 70th minute when the 35-year-old former Socceroo, reinstated to Perth's starting line-up by interim coach Kenny Lowe, made a terrific tackle to deny Adelaide's Marcelo Carrusca what appeared a certain goal.
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