The deciding strike came just a minute form the end of an even first period but after the break the Terras were unable to find a way back into the game, despite a frantic final 20 minutes.
Desperate for a good performance, Stevenage began the match with a flurry of activity, forcing three corners in a frantic two-minute spell as Weymouth struggled to match the home side for pace.
Steve Morison missed a simple chance to add to his tally when he miscued in front of goal with less than 60 seconds on the clock while Dwayne Lee saw a shot deflected wide as the Terras defence was tested to the full.
But as with previous home performances, it soon proved to be an end to end encounter and Weymouth had their chances to go into the lead.
Jason Tindall forced a stunning save from Julian on nine minutes while Lee Elam and Raphael Nade both used their pace to good effect to torment a home defence short on confidence and with Ronnie Henry drafted in at centre-back to replace the suspended Luke Oliver.
The corner count added up for both sides but the pressure wasn't converted into anything of substance, and there were worrying signs for Stevenage as Morison began to look shy in front of goal, drilling one effort tamely at Lee-Barrett on 17 minutes when clear on goal and allowing the keeper to beat him to punch soon after when he looked odds on to connect with a George Boyd throughball.
Weymouth though spurned their own efforts, and ironically their best chance of the first period came through a home blunder. Steve Guppy was at fault as his poor back-pass left Gaia in all sorts of trouble.
His hurried clearance cannon back off Nade but, after the front man forced a super block from Julian, the Weymouth supporters were stunned to see the home keeper race across his goal and stop Elam's goalbound effort on the rebound.
Nade had one more effort before the break but, just as it looked as if it would be a stalemate at the interval, it was Stevenage who snatched the decisive opener.
Boyd, as so often, was the man who crafted the goal, weaving past two challenges and slipping Mark Beard in for a sweet strike past Lee Barrett's dive into the bottom left-hand corner
Ahead, but not in charge, Boro came out for the second period and at last looked confident at the back. Stifled in midfield, Weymouth were unable to create anything of note soon after the restart, and the Henry-Gaia partnership mopped up any threat that came through.
Indeed it was Stevenage that looked set to add the vital second, Boyd twice shooting wide within two minutes of the hour mark as the home side took a grip.
A change of formation from visitors boss Garry Hill did see the Terras enjoy a spell of pressure late on, sub Simon Downer twice coming close in the dying minutes as Weymouth battled to earn a point which their endeavours barely deserved.