The Bristolian wing-back’s curling effort was sandwiched between strikes by forwards Kieran Agard and Aaron Wilbraham as City moved to within six points of promotion.
Swindon, who won the corresponding fixture in November, were only denied an opener by three top-draw saves from goalkeeper Frank Fielding inside the 20th minute.
First the shotstopper denied Andy Williams from an angle and then strike-partner Michael Smith with a fine save to his right.
But the best was yet to come and from the resulting corner the former Derby keeper made a stunning one-handed stop to foil Smith.
Eight minutes later the sold-out Ashton Gate crowd were incensed that a penalty was not given when Raphael Branco looked to have fouled Aden Flint.
The City defender nodded into the side-netting but was furious referee Neil Swarbrick missed Branco’s hands-on defending.
Branco was the guilty culprit for City’s opener against the run of play in the 36th minute. The home side’s high pressing game saw them catch the Brazilian in possession and Agard lifted Luke Freeman’s through ball over Wes Foderingham for his second goal since returning from injury.
Swindon looked for a response and Ben Gladwin was a lick of paint away from a leveller with an effort that deflected onto the crossbar.
Their strikers both missed decent chances to draw Swindon level after the break with Williams firing over at the end of a swift counter-attack and Smith heading wide.
The points were then sealed in style when Bryan looked up and bent a left-footed effort into the top right-hand corner from the edge of the box.
They added gloss to the scoreline with three minutes to go when Freeman’s pull-back flew across the box for captain Wilbraham to drill into the bottom corner.