Steven Schumacher's 10th goal of the season after 21 minutes separated the sides, but Swindon failed to muster a shot on target until late in the second half when Luke Rooney's free-kick forced Bury goalkeeper Trevor Carson into action.
Shakers boss Blackwell admitted he had not been surprised by the way his lowly outfit had taken control of a match they had expected to lose.
"This was a fantastic performance," he said "The lads have shown what they can do and we deserved the points.
"We stuck to our game plan against one of the top teams in the league. Against Notts County recently we paid for a poor 10 minutes.
"But where we failed at Notts County, we succeeded against Swindon.
"This was a massive win for us, but anyone who has seen us recently would not have been surprised.
"Craig Fagan was awesome and led the line brilliantly from start to finish. A great performance.
"But there is still plenty of work to be done. We need to make changes and to bring in more players, but in fairness to the players already here we have been playing well for a long time.
"I watched Swindon several times lately and to get a win against them is tremendous.
"Overall I thought we outplayed and outfought a good Swindon side, so there is a lot to be proud of."