The Ghana international struck after nine minutes and, after Steed Malbranque had missed a first-half penalty, added a second four minutes from time to help ease the pain of last weekend's 5-1 humiliation by Newcastle.
But in the meantime, Stoke were denied what they felt was a certain penalty for handball after Lee Cattermole smuggled Kenwyne Jones' header off the line and then had skipper Ryan Shawcross sent off for two bookable offences inside three minutes.
It was the visitors' fifth successive defeat in all competitions and fourth in the Premier League, and it proved the perfect tonic for their hosts after a difficult week.
The fall-out from last Sunday's derby drubbing at Newcastle has been uncomfortable and intense despite heartfelt apologies from Steve Bruce, chairman Niall Quinn and leading scorer Darren Bent, and the Black Cats ran out knowing the repair job had to start in earnest this afternoon.
Having kept his big-money summer signing in reserve for much of the time since his arrival, Bruce had revealed in advance that Gyan would start today, although his hopes of playing him alongside Bent were dashed by the England international's unavailability.
However, it took the Ghanaian just nine minutes to make his mark when he was presented with a tap-in after keeper Asmir Begovic had parried Nedum Onuoha's initial effort.
The horror of their black day on Tyneside seemed to have been exorcised as Sunderland set about the task of rediscovering the form which had seen them put together a seven-game unbeaten league run before the debacle at St James' Park.
But where Newcastle had been clinical six days ago when presented with the chances which enabled them to establish a 3-0 half-time lead, the Black Cats were guilty of squandering a glorious opportunity to put themselves firmly in the driving seat.
Malbranque's intelligent pass put Gyan in on goal 17 minutes before the break and, in his attempts to get goal-side of the striker, Jon Walters hauled him to the ground.
Referee Martin Atkinson had little hesitation in pointing to the spot and with Bent absent, Malbranque stepped up to take the penalty.
He side-footed his shot low to Begovic's left, but the keeper guessed correctly and pulled off a vital save to keep the visitors in the game.
Shawcross made a good block to deny Malbranque with four minutes of the half remaining, and Bruce's men headed for the dressing room hoping Malbranque's miss did not come back to haunt them.
For their part, Stoke offered little as an attacking force with Sunderland old boy Jones seeing too little of the ball to do any damage, although keeper Simon Mignolet managed to block the Trinidad and Tobago international's 12th-minute toe-poke with his foot.
The visitors had made little use of their prowess from set-plays, but almost did so to good effect within three minutes of the restart.
Robert Huth's long throw was headed on by Walters at the near post and Matthew Etherington arrived on cue to smash a left-foot shot towards goal, were it was blocked bravely by Michael Turner.
Potters boss Tony Pulis made his move with 57 minutes gone when he withdrew Walters and replaced him with Ricardo Fuller, who joined Jones in attack and Tuncay Sanli slipped in behind them.
They went close with 61 minutes gone when keeper Mignolet failed to collect Etherington's left-wing corner and first Huth and then Tuncay attempted to pounce on the loose ball, only for Phil Bardsley to get there first.
Sunderland defender John Mensah, who had earlier twice gone down in apparent agony clutching his shoulder, was replaced by Anton Ferdinand with 23 minutes remaining as Bruce attempted to strengthen his rearguard for weathering an expected late onslaught.
But his side enjoyed a major escape with 19 minutes remaining when Jones' header from an Etherington corner looked to be creeping inside the far post until Cattermole bundled it away with more than a hint of handball.
The Black Cats skipper hacked another Jones header off the line from the resulting corner to deepen Stoke's woes with the pressure mounting.
Shawcross, who had earlier been booked for a dreadful challenge on Danny Welbeck, was cautioned again, perhaps harshly, for a trip on Gyan and was sent off eight minutes from time.
And the points were safe when Gyan fired handsomely into the bottom corner from substitute Kieran Richardson's left-wing cross.