Ritchie de Laet had scored against his old club to put the Canaries in front at the break.
After Jon Walters saw his penalty saved following Leon Barnett's red card, it looked as if Norwich would hold on, only for Stoke's pressure to eventually tell with Jones nodding in from close range.
Anthony Pilkington - one of half a dozen changes by Canaries boss Paul Lambert from the side which drew 1-1 at Wigan - shaped to shoot from 25 yards, but could only blast the ball straight at Robert Huth.
The visitors were forced into a change on the half-hour when Jermaine Pennant, struggling with what looked like a calf problem, was replaced by Danny Pugh.
Jonathan Woodgate went into the book after 36 minutes for upending Elliott Bennett out on the far touchline.
From the resulting free-kick Norwich took the lead when former Stoke defender De Laet, on loan from Manchester United, glanced Bradley Johnson's curling delivery into the far corner.
After 57 minutes, Grant Holt was penalised for pushing over Glenn Whelan to concede a free-kick in a dangerous central position, some 25 yards out. Matthew Etherington beat the wall, but John Ruddy flicked the ball over the crossbar.
Disaster struck for the home side on 63 minutes when Walters got ahead of Barnett and into the Norwich box, before tumbling. Referee Neil Swarbrick ruled Barnett the last man, awarding a penalty and then sending the Norwich defender off. Walters took the spot-kick himself, but Ruddy guessed correctly to make a fine save.
Jones blasted over from close range, but the visitors' pressure finally told in the third minute of added time when the striker headed in Whelan's cross.
Source: PA
Source: PA