Burgess added a fourth goal in the penalty shoot-out but could only huddle next to his team-mates and watch from halfway as Bay Olympic keeper Gareth Juddery dived high to his left and deflected Dunedin's fourth shot wide.
Defender Julyan Collett then sealed the win with a clinical strike - the Aucklanders winning the penalty shoot out 5-3 after the scores were locked 3-3 at the end of extra time.
For 120 minutes the home side battled a talented Bay Olympic in bitterly cold conditions, desperately trying to keep its Chatham Cup dreams alive.
The drama unfolded on a patchy blanket of snow and hail which covered the Caledonian ground in front of about 100 brave souls.
Wind and sleet swept in from the south, battering the players and hiding the ball in dull skies. It was more like torture than a quarterfinal and neither side wanted to emerge from the ordeal defeated.
Thwarted, yes, but far from disheartened, Burgess told the Otago Daily Times he was "really proud of the lads".
"There is not much you can do in penalty shoot outs," he shrugged.
"It is a lottery and I told the boys 'whatever happens, happens. If we miss one, we miss one'. And it happened."
"We're absolutely disappointed with the result but the boys gave it everything they had. They were tough conditions and we were up against a very, very good side."
From the outset, Bay Olympic looked the more accomplished side. Ross McKenzie threaded some wonderful passes through to his strikers and remained a constant threat.
The visitors looked at their best from set plays with fellow midfielder Jake Butler forcing the first save of the game.
Dunedin Technical absorbed the pressure but was not able to create any momentum on attack. Nothing was dropping for Burgess up front and it just seemed a matter of time before Bay found a goal.
It should have come early in the second spell. McKenzie made an inch-perfect pass which found Colin Gardyne on the right flank. He whipped a cross back into an unmarked Robbie Greenhalgh. With just the keeper to beat, Greenhalgh cracked it straight at gloveman but got a second opportunity when the deflection fell at his feet. Somehow, Michael Walker, who was flat on his back, thrust out a limb and knocked it wide.
Brilliant desperation from the keeper but poor execution, as well. It was the first of two or three good goal-scoring opportunities which were squandered.
Just when it seemed like Technical was destined to hold on, the referee found a penalty out of a seemingly innocuous challenge in the box. McKenzie converted what, for a long time, appeared to be the winning goal.
That was until a cross was flicked forward and found Burgess hovering in the danger zone. His precise strike sent the game into extra time.
Then Burgess scored the goal of the match, a blinding strike from 30m. The keeper made good ground to his left but could only fetch it from the net and shake his head in appreciation.
Joy turned to despair when defender Matt Cunneen poked the ball in after a goal mouth scramble to tied the game 2-2. But there was still time for a few more twists.
Burgess completed his hat trick when he was brought down in the box and converted from the penalty spot. But Dunedin was robbed of the glory when Collett smacked in a stunner from about 30m.
The news was much better for fellow Dunedin club Caversham. The South Dunedin club has reached the semifinals thanks to a 4-0 against Onehunga at Waikaraka Park in Auckland yesterday.
Four red cards were issued in the game. Caversham finished the game with 10 players while its opponent had just eight players remaining.
Patrick Fleming opened the scoring from the penalty spot, Tom Jackson scored twice and Ant Hancock was also on the score sheet.
In the other two quarterfinals, Napier City Rovers beat Manurewa 3-2 and Wairarapa United overcame Waitakere City 4-1.
The semifinal draw will be conducted tomorrow.
