But the home side were caught cold in the 68th minute as Storey's cross-cum-shot looped over goalkeeper Owain fon Williams to send Tranmere sliding to a first home defeat in four games.
The Robins' opener came as Smith lashed home his seventh of the season via the crossbar from the edge of the area.
But just three minutes later Tranmere levelled when Power got his first of the season by steering Liam Ridehalgh's cross into the top corner.
Taylor's spectacular overhead kick went inches wide before Swindon goalkeeper Wes Foderingham made a superb save to deny Cassidy from close range.
Lowe was then denied a stunning 19th of the season by the crossbar after he controlled Taylor's pass and attempted an acute lob.
After Taylor had smashed another shot against the crossbar, Rovers were caught by a Swindon sucker-punch as Storey's fortuitous strike handed the visitors all three points.
- Sports & Recreation
- Ash Taylor
- Jake Cassidy
- Alex Smith