The Sulphurites dominated possession for long periods and passed the ball well, but were unable to find a way past their hosts' stubborn five-man defence.
Having been stung on the counter-attack seven minutes into the second period, Town couldn't force their way back into the contest and their three-match winning run ground to a halt.
Weaver, however, was able to take plenty of positives from what he saw at Broadhall Way.
"I thought it was a really good performance and some of the best football we have played this season," the Harrogate boss reflected.
"We were fluid, passed the ball well, created a lot of problems. Yes, we lacked a little bit of end product but sometimes you don't get the rub of the green.
"The effort throughout was brilliant. We have played a lot worse and won games. We played through the thirds as well as we have. It wasn't just about being direct and winning the war. They all played their part and there weren't really too many faults to pick on, but sometimes it doesn't go your way in football.
"One long ball over the top did us, it's unfortunate, but the lad took his chance really well. I couldn't really fault the performance. We've had the lion's share of possession and the players should take a lot from the game in spite of the result.
"If we play like that from now until the end of the season it is going to be an exciting time."
Town fell behind on 52 minutes after Ryan Fallowfield made a mess of dealing with Jamie Cumming's long clearance, allowing Stevenage forward Elliot List to run clean through on goal and beat a horribly exposed James Belshaw.
Having gone close to breaking the deadlock in the first period through Josh McPake, George Thomson and Lloyd Kerry, the visitors created two good second-half chances to get themselves back into the contest.
First, Jack Muldoon got on the end of Mark Beck's flick-on, only for the 31-year-old to fire over the bar from close range, then Warren Burrell missed the target with a free header from Thomson's free-kick.
This left Weaver, who felt his team benefited from having a quality playing surface to knock the ball about on, to rue a lack of clinical edge in the final third.
"It was a very good pitch to play on, very firm, one of the best pitches in the league," he added.
"Their tactics allowed our full-backs to get on the ball and we built it from the back. The two centre-halves deserve a lot of credit for collecting it from the keeper. Tactically, we looked at it and thought we can build it from the back, There was room for us to play.
"Once they got their goal, they did retreat and it was like a 4-5-1 against us and hard to prise open. The football that we played was really good on the eye but obviously you're there to be shot at because we haven't scored a goal, and that's what was missing.
"We've just got to show a little bit more composure in those moments to get the ball over the line and that's all that was missing today.
"Unfortunately we've come away losing the game, but we're not too downhearted. We are five points off the play-offs which is unreal compared to eight weeks ago, let's not beat ourselves up."
Harrogate remain 11th in the League Two standings despite Saturday's loss.