The Sulphurites headed into the fixture having failed to serve up a victory on home soil for two-and-a-half months and off the back of a run of eight defeats in 10 matches.
But, having spent time interacting with the Wetherby Road faithful before kick-off as part of the club's 'Supporter Thank You' event, Weaver said his players were visibly lifted by the positivity which radiated from the terraces.
And Town's manager believes that his team "fed off" that energy as they went out and delivered their most complete performance at the EnviroVent Stadium in some time.
"It was a good atmosphere in the ground before the game," he said.
"We were trying to thank the fans for their continued support. It's not been an easy season by any stretch, particularly in recent months, but it was fantastic to hear the comments from the supporters and we know that their support is growing and it is resounding.
"It let us know that they're still with us and that's what we needed. I think the feel-good factor was here before kick-off and it filtered through to the players, that positivity, and then they've turned it on in style.
"Young and old, they all seemed to be really behind us and sometimes you don't realise that, you just go home not knowing. There's been a lot of pressure on us and a lot of things go wrong with injuries and big players out, so every hit has been hard to take, but today was a good day for the club with the camaraderie between the fans and the players.
"It made the place feel a lot more together. We fed off that positivity and you saw that in the performance. We could have scored seven or eight goals."
Given their recent poor run, which had seen them lose four games on the bounce prior to beating Carlisle, Weaver admitted that he was not sure how Town's fans would react to the 'Supporter Thank You' event.
But, the Sulphurites chief revealed that he was pleasantly surprised by the warm reception that he, his players and staff all received.
"The fans have turned out and, to be honest, I was a bit sheepish about it because I thought 'I'm not sure they'll be thanking us at the minute, or certainly not thanking me', but they were brilliant," Weaver added.
"It's been hard for everybody in the building, we are not used to having a sequence of defeats. It's probably the first time for quite some time, but the fans, what they were saying on the terraces before the game when the lads were signing autographs, it was so positive and I certainly got lifted by it.
"I said to Rachel 'that's lifted me, I didn't even think they'd want me over there'. I wasn't going to go anywhere near it because I didn't think I'd be wanted, but someone shouted me over so I did. I know that I certainly felt better because you get to hear those positives.
"I think it made the place feel a lot more together. The chemistry was there with the players and I think that events like that, the more the merrier for me because we need the fans, it's vital, we can't keep sustaining things, never mind growing if we don't have them with us."
Saturday's victory lifts Harrogate one pace to 19th in the League Two standings with two games remaining this term.