While Burns was forced to take a spot-kick twice to secure his hat-trick, the 26-year-old forward slotted both efforts to be eventually awarded his third goal, which capped off Wellington's impressive home win.
The Phoenix had lost on successive road trip but after a nervous start in the New Zealand capital, Ernie Merrick's men blitzed City to move six points clear of the Melbourne-based side.
Burns opened the scoring in the 33rd minute before Roly Bonevacia struck five minutes before the break to make it 2-0.
Just two minutes into the second period and Burns showed real poise in the box to extend Wellington's lead further, while a Roy Krishna bullet in the 50th minute effectively ended the contest.
David Williams pulled a goal back for City in the 68th minute with a penalty but the Phoenix would be awarded a spot-kick of their own with five minutes remaining.
Burns - who missed a one-on-one when on a hat-trick earlier in the season - converted, before being forced to take the penalty again for encroachment, but he kept his nerve to again score and make Phoenix history.
The goals took Burns to the top of the A-League goal-scoring charts with seven, two ahead of Melbourne Victory's Besart Berisha and Andy Keogh of Perth Glory.
Elsewhere on Sunday, Newcastle Jets slumped to their worst start to an A-League season after being held 1-1 by Central Coast Mariners at Hunters Park.
Edson Montano's scrappy effort early in the second half had the Jets set for their first league win of the campaign.
However, Richard Vernes got ahead of his marker to convert Michael Neill's cross and snatch a point which does little to kick-start the Mariners' season in truth.
Both sides hit the woodwork in an attempt to snatch the spoils, but both remain firmly rooted in a group of five sides looking likely to fight it out over the final play-off spot in the A-League.
