The League Two side had looked dead and buried after only 28 minutes of the tie when Craig James saw red early on and Vincent Pericard had put the hosts in front.
But a 30-yard effort from Carlos Logan and Joachim's second half screamer gave Darlington there first win over the Potters in five previous attempts.
If travelling to Championship opposition wasn't tough enough, an early rush of blood from James saw the Quakers facing a real uphill struggle.
James got his marching orders in the 12th minute, when, with the ball already out of touch, Kevin Harper was left chewing turf following a shove in the back from the full-back.
Stoke immediately set about making the most of their numerical advantage after Darlington had had the best of the early sparring.
Visiting skipper Patrick Collins thought he had got his troops a dream start after only four minutes only to see his bullet header come back off the upright.
But with a man over, City quickly set about asserting their Championship credentials through a 28th minute clinical finish from Pericard. The French forward showed bags of composure in front of goal and stroked the ball into the bottom corner after latching on to a wonderful through from Carl Dickinson.
The hosts thought that Pericard's early strike had put them in the comfort zone but the plucky visitors had other ideas and were back on level terms just before the interval.
Logan hit Steve Simonsen's bottom corner from 30 yards after the fleet footed winger was allowed to skip in from the left flank with some ease.
And, with the second half only eight minutes old, Stoke's casual approach was exploited again as Darlington took a deserved lead through a wonder strike from Joachim.
Jeff Whitley, making his home debut, was caught in possession and Joachim took full advantage and smashed the ball into the roof of the net past the flapping City keeper.
There may have only been just over 3,000 home fans but they were certainly making there feelings on trailing to lesser opposition felt as City pressed for an equaliser.
But it was the travelling fans who had the most to sing about when the final whistle eventually concluded a nail-biting six minutes of injury time.