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Louis Fenton saw his strike hit the side-netting shortly after the hour as the hosts pushed for an opener, while Roy Krishna made an immediate impact off the bench to force Ante Covic into making a solid save.
Wellington finally made the breakthrough 12 minutes from time when Bonevacia released Krishna down the left, with the resulting cross into the area being diverted home by Burns.
Michael McGlinchey was gifted the chance to double his side's advantage from the spot shortly after when Burns was brought down, but Covic kept out the penalty to keep his side in the contest.
The Wanderers were unable to find a way through in the remaining time, however, ensuring that the Nix claimed all three points which sees them climb up to fourth place in the table.