There were plenty of things to celebrate from this Phoenix performance, which came on the back of consecutive away defeats to Adelaide and Perth.
The result kept them comfortably in fifth place on the A-League ladder and they will look to push towards the top four with an away trip to meet the lowly Jets in Newcastle on Saturday.
Wellington were nervous to begin with against City, and the visitors, who have struggled this season despite beginning the campaign as one of the title favourites, should have found their way on to the scoreboard during the early stages.
"You can't control and dominate a game for 90 minutes," Phoenix coach Ernie Merrick said. "The other team's always going to have a shot at goal, that's why you've got a defence and that's why you have a goalkeeper."
Burns broke the deadlock in the 34th minute after he swooped on a wayward back pass from Melbourne's Erik Paartalu to put the home side ahead.
"After that, we lost control," City coach John van 't Schip said.
From there, Wellington surged as Burns added a second, while Roly Bonevacia and Roy Krishna also scored and by the 50th minute, the Phoenix had a 4-0 lead.
David Williams later pulled one back for City with a well-taken penalty but they never threatened a comeback. Burns then had the final say as his third goal created a slice of history for the Phoenix.
Burns, who is leading the A-League's golden boot standings with seven goals for the season, is making a strong case for a recall to the Australian national side for next month's Asian Cup.
His combination up front with Michael McGlinchey has given the Phoenix an attacking edge that has never been seen in Wellington and his move to the capital has revived his career.
The death of Australian cricketer Phillip Hughes has been widely felt across many sporting codes and football was no different as the 7410-strong crowd stood and applauded during the 63rd minute of yesterday's game as a mark of respect to the batsman.
Burns, who grew up in New South Wales, even played a few schoolboy cricket matches alongside Hughes.
"It was quite a sad week in sport in Australia," Burns said.
Wellington Phoenix 5 (Nathan Burns 34, 47, 85 Roly Bonevacia 40, Roy Krishna 50) Melbourne City 1 (David Williams 68) halftime: 2-0