City coach Karl Hayward had called for more passion from his team and he got it in spades, as his players grafted for 120 minutes to deny a quality side with a number of New Zealand academy players in their ranks.
While the visitors from Wellington shaded most of the 90 minutes, looking the more composed and dangerous side, City kept their shape, defended doggedly and kept trying to play football.
Gradually, they began to assert themselves and their battling attitude got its reward in the first period of extra time.
Chris Annabell, who had toiled willingly as a lone striker, gamely gave chase to a long, looping volley from Mike Hewitt.
It looked like a lost cause as United defender Kyle Segebart played it back to keeper Andrew Withers but Annabell persisted and, when the keeper dwelt on the ball too long, he got a foot in.
The ball ricocheted toward the edge of the penalty area, but a somewhat panicked Withers clumsily grappled Annabell to the ground and the ref pointed to the spot amid howls of protest.
Graham-Luke became Cool Hand Luke as he drove it low and hard inside Withers' right-hand post.
City had 15 nerve-racking minutes to hold on, but they did so with some comfort, with headers from Nick Edginton and Caleb Hilbron going high and wide, and keeper Hayden Englefield making a fine save from Riley Kelliher.
Former Wanganui Athletic keeper Englefield, only signed recently and covering for the injured Nick Hayward, looks a good short-term acquisition and he pulled off a number of good stops.
But the best save of the match was by centre-back Ben Toyne who came to the rescue when a Hilbron shot was rolling toward the net with Englefield stranded out of position. A desperate slide by Toyne saw him hook the ball away inches from the line - a perfect example of the commitment that carried City to victory.
The game ended with some drama. As United became more desperate, Hewitt pushed forward and nearly made the game safe with a second goal. Then he went to take a throw-in and punched the ball away, a piece of silliness that earned him a yellow card for time-wasting and, with one he had received earlier, saw him sent off.
As the fulltime whistle blew, United's Cory Brown voiced his displeasure to the referee and he too was red-carded. City, unbeaten in the Western Premiership this term, forward to a third-round Chatham clash on June 23.