The visitors can have no argument with the outcome of a game which Stevenage dominated from the outset, creating a host of gilt-edged chances.
With Exeter pegged back early on it was Stevenage who applied the pressure with Jon Nurse seeing an early header saved, while Simon Weatherstone fired just wide after Anthony Elding had earlier guided a header across the face of goal.
Man of the Match Boyd orchestrated the Boro attacks and Exeter had no answer to his polished performance, which saw further chances fall to Nurse and Barry Laker before the break.
Exeter brought on the industrious Tony Scully at half-time, but though he upped the tempo for the visitors, it was still Stevenage who looked more likely to score.
An opener looked on the cards when Stevenage were awarded a penalty on 68 minutes after Chris Todd handled a Nurse cross.
Elding stepped up and drilled the ball wide of the keeper's right-hand post but it signalled the start of a Boro attacking spree which led to the first goal just two minutes later.
Boyd proved the inevitable scorer as he picked the ball up in the Exeter half and evaded two challenges before firing past Paul Jones an unstoppable drive.
Jon Challinor and Lee Phillips both had belated efforts for Exeter but it was Stevenage who dominated in the closing stages and grabbed a deserved second two minutes from time, when Boyd delivered a simple through ball which allowed Dannie Bulman to run through and fire home.