So complete was the home side's dominance, that Exeter managed just four attempts on target and from those, they put two in in the dying seconds of the first half.
But Stevenage began the game in impressive fashion and could have gone ahead after six minutes, when Anthony Elding rounded the final defender only to see his drive flash wide across goal.
Further chances fell to Jason Goodliffe, Dino Maamria and Jon Brady before Stevenage got the goal their pressure deserved on 34 minutes.
Elding supplied the finishing touch with a pinpoint header into the net and could have made it two, just three minutes later with a cheeky 40-yard lob which caught keeper Paul Jones off his line, but it sailed just over.
Seemingly heading for a half time deficit, Exeter amazingly turned it round in added time when first Steve Flack turned and fired them level inside the area, before the same player latched on to a neat Gareth Sheldon throughball before driving low past home keeper Andy Woodman.
Stevenage however were not shell shocked at the restart and continued in the same impressive vein with Dannie Bulman and Maamria both coming close before a powerful Maamria header from a George Boyd free-kick on the hour saw the home side pull level.
What then followed was half an hour of constant pressure as they peppered the goal and even hit the woodwork through Simon Weatherstone, before, with just five minutes remaining, Maamria produced an acrobatic effort to send the ball in and secure three richly deserved points.