The Sulphurites looked on course for a fourth League Two victory in as many games, leading 1-0 at the interval courtesy of Jack Muldoon,s 43rd-minute penalty.
But the Grecians, who were reduced to 10 men when Harry Kite was shown a second yellow card in the process of conceding that spot-kick, levelled late on through Josh Key.
“As a collective, we weren’t quite at it. We didn’t show that vitality in front of goal or defensively," Weaver reflected.
“It’s a real high-intensity game we play when we’re on it. Maybe coming back [from Leyton Orient] at three or four in the morning on Wednesday might be a factor, but we’ve got to rest up and go again.
“I don’t think it’s a lack of effort, I think it’s just running out of steam - on this occasion - and it’s something that we’ll look at in our recovery and hopefully we’ll get that edge back about our game for Tuesday.
“We don’t want to be that team who win three then lose three. You have to take a point sometimes when you’re not at your best and then move on, be a little bit annoyed with ourselves and come out firing for the next game. Hopefully this will fire us on to another win."
Town were second best before falling behind, then failed to really press home their numerical advantage during the second period, with Weaver conceding that his players didn't do enough to put the game out of Exeter's reach.
But they finish the weekend second in the table and remain unbeaten this term.
“I don’t think we really deserved the three points," he added.
“Exeter performed well. I thought that we were second too many times during the first half. We didn’t win the first or second ball enough to create an energy or momentum in the game.
“We didn’t pass the ball as much as we would have liked when they went down to 10 men. When we did, we caused them problems but we didn’t do it enough to really take charge of the game. So, I think it was a fair result.
"But, we’re very proud of the start we’ve made. All in all, we’d have accepted being second and two points off the top of the league after four games.”