An intricate free-kick produced the first chance on nine minutes when Kevin Harper side-stepped an onrushing defender before blasting a 25-yard shot just over Rami Shaaban's bar.
Darel Russell was a force in the midfield and combined well with Harper down the right to produce a host of chances throughout the afternoon.
Harper chipped his cross from the right by-line in the 23rd minute but Gifton Noel-Williams failed to get a clean head on the ball.
Stoke's continued pressure finally paid off in the 39th minute when new signing Michael Duberry was brought down by Mark McCammon who conceded a penalty.
Noel-Williams stepped up and stroked the ball neatly into Shaaban's top right corner.
Only three minutes later, Daniel Harding handed the Potters a second penalty when he was adjudged to have held back Gerry Taggart.
Yet again Noel-Williams supplied the finishing touch with a carbon copy of his previous spot kick.
The Seagulls stepped up their efforts after the break with the addition of Nathan Jones increasing their attacking options.
Steve Simonsen was called into action for the first time on 56 minutes when Jones' 20-yard strike required a finger-tip save to push the ball over the bar.
Simonsen was again in the thick of things on 78 minutes when another long range shot, this time from Gary Hart, had to be parried at full stretch.
The visitors had a late call for a penalty following a push by Duberry but City comfortably secured an important three points.