Arsene Wenger's side never got to grips with the aerial power of their hosts and, despite bags of possession, they hardly tested home goalkeeper Thomas Sorensen.
Arsenal bossed the early stages, but that meant nothing when fine play between Mamady Sidibe and Ricardo Fuller led to Delap's first long throw of the match in the 11th minute.
Delap's wind-assisted delivery was huge and it only required the slightest of touches from the head of Fuller before the ball flashed inside Manuel Almunia's right-hand post.
The goal gave Stoke huge belief and Arsenal appeared shell-shocked as the underdogs pressed hard for a second.
Stoke almost got it in the 20th minute when Delap looped a header over the stranded Almunia and Gael Clichy had to clear the ball off the line.
Arsenal finally landed a blow of note five minutes later when Emmanuel Adabeyor's 20-yard shot was held low down to his left by Sorensen.
And a magnificent pass from Cesc Fabregas deserved better than a poor chest control by Nicklas Bendtner, which allowed Sorensen to claim the loose ball.
But Stoke, well organised and biting into their challenges, were largely untroubled and continued to give their illustrious visitors problems at the other end of the field.
The Arsenal defence was struggling to deal with Sidibe, in particular, with City making the most of their height and weight advantage.
Arsenal, no doubt given a stern half-time talking-to by Wenger, upped the ante after the interval with Fabregas pulling most of the strings.
But Stoke remained a tough nut to crack and Wenger made his first change in the 58th minute with Theo Walcott brought on in place of Bacary Sagna.
The young England star won a corner from his first touch and Kolo Toure almost levelled when he volleyed Fabregas' delivery a foot too high.
Robin Van Persie soon followed Walcott into the attack, with Denilson making way, as Wenger went for broke.
But Arsenal were left with a mountain to climb in the 73rd minute when another Delap throw was flicked on by Ryan Shawcross and Seyi Olofinjana bundled in the loose ball.
And it was as good as over three minutes later when Van Persie was sent shown a straight red card for a ridiculous kick and barge on Sorensen.
But there was a sad ending to the game when a poor challenge by Delap resulted in Walcott being stretchered from the field with a leg injury in stoppage time.
Clichy scored with long range shot which took two deflections from Fabregas' short free-kick, but it proved only a consolation as Stoke ran down the clock against nine men.