Livewire forwards Green and Constable helped make a goal for each other in the first half, and Potter came off the bench to settle the nerves of nearly 35,000 Oxford fans in the final minute of a pulsating Blue Square Play-Off final.
Those supporters had bitten their nails to the quick after seeing York, gifted a goal by U's goalkeeper Ryan Clarke, push Chris Wilder's men back and back in the second half.
But it was a rousing performance on the greatest stage of all which took Oxford United back to the Football League after a four-year absence.
On an afternoon of sunshine and heavy showers at Wembley Stadium yesterday, this final brought more than 33,000 Oxford fans an evening of appropriately fluctuating emotions and nervous worry.
Green gave the U's the lead in the 15th minute and it was both a brilliant finish and good build-up play in the lead-up to it.
Constable challenged with goalkeeper Michael Ingham, which forced the ball loose, Jack Midson then produced a brave header to knock it sideways, and Green took the ball on his chest and fired in left-footed, high into the unguarded goal.
If the Oxford fans were jubilant then, they were in dreamland six minutes later as Constable doubled the lead.
This time Green set him up, Constable's first touch giving him the perfect angle for a left-footed drive - which he hammered low into the net.
Just as in the semi-final second leg against Rushden, it had been two goals in rapid succession, and York were left rattled.
After that the U's played some terrific football, one mesmerising move in the heavy rain ending with another Green shot that Ingham turned around his post with some difficulty.
United were looking comfortable, until a bad error by Supporters' Player of the Year Clarke three minutes before half-time gave York a lifeline.
Alex Lawless's up-and-under right-wing cross should have been meat and drink to him, but the Oxford keeper let the slippery ball through his grasp and dropped it over the goalline.
That goal-out-of-nothing gave the Minstermen a massive boost and it was clear they had belief again in the minutes before half-time, and from then on through the second half.
However, some strong and fiercely determined defending, and great blocks from Mark Creighton, Jake Wright and Anthony Tonkin, kept City out.
And then Potter applied the coup de grace after exchanging passes with fellow sub Sam Deering on a swift counter-attack.
From the moment the players emerged from the tunnel to a deafening crescendo of noise, and flames shot 40 feet into the air as part of the Wembley welcome, it was clear that this was to be like no ordinary occasion for Conference teams more used to venues like Altrincham and Grays.
When the players were introduced to the dignitaries, including Sir Geoff Hurst, the rain was hammering down, yet it would take more than that to drown out the noise of the 30-000 Oxford fans, who turned one half of the stadium into a sea of yellow.
The other end was only sparsely populated by York supporters, though they too were doing their best to make themselves heard to lift their team.
United began well, with Matt Green forcing a throw-in deep into York territory, and then Damian Batt's long throw causing pronlems inside the opening minute.
And if the defence needed something to calm them down, that followed moments later when Richard Brodie tried to outsprint Jake Wright along the right flank, but could only do so by fouling.
The surface was skiddy after the rain, and Ryan Clarke had to have his wits about him to backpeddle and then push the ball over his bar when Chris Carruthers knocked the ball back in .
Jack Midson's chipped pass released Matt Green, who reached the ball just ahead of keeper Michael Ingham but in the challenge, it ran free.
Simon Clist retrieved it, but although Ingham was way out of his goal, he was unable to get it back into the box where two teammates were waiting.
On the break, Alex Lawless's cross was just too high for Michael Rankine, before Chris Carruthers managed the game's first effort on target with a low drive that brought a save from Clarke on his knees.
Space opened up for Adam Chapman to try his luck at the other end a few minutes later, but he sliced his 25-yard shot.
Oxford's two goals in six minutes put them in charge of the match, and they intelligently then began to keep possession whenever they coluld, with neat interchanges all across the pitch.
Another bright attack gave half a chance to Green, as Midson dummied, and the on-loan Torquay striker shot on the turn, but it deflected for an easy save for Ingham.
Midson missed a glorious chance to make it 3-0 when he headed against the left post midway through the half, when it had seemed easier to score because the goal was at his mercy.
Oxford had dominated possession in the opening half hour but York started to come back into it.
Lawless was booked for a late challenge on Anthony Tonkin and then Jake Wright, who had been immense at the back, made a brilliantly-timed tackle on Brodie.
It was all going perfectly for United, but Clarke's uncharacteristic mistake left the game back in the balance.
But United wre back on the front foot from the very start of the second half, a Wright header shepherded around his left post for a corner by a defender.
A bad mistake by Tonkin let in Michael Rankine in a great position nine minutes after the break.
But, fortunately, for him, and United, Rankine shot wide.
It was becoming a very open match now, and probably too much so for Oxford, as York's front two were causing problems, and the game became stretched.
City midfielder Neil Barrett sent in angled 18-yard shot, that Clarke got down well to save.
Oxford's players were similarly willing to try a shot whenever they could. Simon Clist cut in from the left and struck a sweet shot that Ingham pushed up and away, and Chapman sliced the follow-up some way wide.
There must have been a few hearts in mouths among the United fans when another difficult cross went Clarke's way midway through the second half. But this time he took it expertly, his confidence seemingly restored.
Both managers made changes for the games's last quarter, Chris Wilder senind on Alfie Potter soon after Green had taken a knock in a fierce challenge, and York boss Martin Foyle throwing on extra attackers Michael Gash and Courtney Pitt.
Then Sam Deering came on to tighten united's midfield further, giving them four in the centre of the park, and with Deering operating behind Constable in a 4-4-1-1 formation.
There was so much nervous tension among the Oxford fans that the singing had stopped. Most of them were just willing the final whistle to go.
Some Potter magic helped earn a corner near the end, and when Deering's trickery, and Bulman, then brought another flag-kic, it was entirely understandable that they played it short to run down the clock.
A brilliant run by Potter, inside then out, then in again, should have brought a third goal for Oxford. He laid the ball to his left, but Constable lofted his shot over the bar.
Two minutes from time, Clarke caught a difficult cross under extreme pressure.
It was all York in the last few moments.
But then subs Potter and Deering combined beautifully on a fast break out of defence. Potter passed to Deering on his right, Deering returned the favour, and Potter shot low past Ingham for another cracking finish - and a glorious end to what, for everyone supporting Oxford will testify, was a momentous match, and an end to a decade of hurt.