The home side took the lead in the 35th minute when Burns raced onto a through ball and managed to round Andrew Redmayne, before slotting the ball into an empty net.
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Wellington doubled their lead five minutes later when Michael McGlinchey released the run of Roley Bonevacia, and he powered the ball past Redmayne from close range.
The Phoenix could have gone three goals ahead just before the break, but Redmayne made a good stop to deny Roy Krishna from finding the net with a headed effort.
In the second half, Burns grabbed his second goal of the game in the 47th minute when he rifled the ball into the bottom corner with a ferocious strike from the edge of the area.
Krishna put his side out of sight when he netted the fourth of the match as he latched onto Tom Doyle's ball into the box and buried past Redmayne.
Burns made Wellington history by becoming the first player to score a hat-trick for the club when he found the net from the spot late in the game, following a foul by Rob Wielaert, as the home side romped to victory.