Despite dominating the opening exchanges the home side were unable to turn their advantage into points and it was left to skipper Chris Paterson to open the scores with a penalty.
The score appeared to steady Edinburgh and they cemented their lead with the first try of the game after 22 minutes.
Winger Simon Webster sparked the move and also got on the end of the flowing attack, after some good work from Marcus Di Rollo, to dot down for the score.
Connacht rallied with a penalty from Paul Warwick and then the game threatened to turn ugly before the break as tempers flared.
The result was Edinburgh breakaway forward Ally Hogg being sent to the bin for a high challenge and Connacht lock Christian Short followed him into the sin-bin for an illegal block on Francisco Leonelli.
After the break Edinburgh were handed a penalty try when Paterson was held back as he looked to chase down a chip ahead from Leonelli.
Edinburgh put their foot on the accelerator later in the half with three tries in a few minute and in doing so bagging the bonus point.
Firstly Hogg ploughed over in the corner before Dewey finished off a searing move featuring Paterson and the impressive Leonelli.
Paterson rounded of proceedings by collecting a pinpoint pass by Matt Mustchin to score the final try of the game.
|Edinburgh Gunners Score Card|
|Connacht Rugby Score Card|