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 after 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,” said Weaver.
“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.”
Town were second best before falling behind, then failed to press home their numerical advantage during the second period, with Weaver conceding his players didn’t do enough to merit all three points.
“We didn’t pass the ball as much as we would have liked when they went to 10 men,” he added. “When we did, we caused them problems but we didn’t do it enough to really take charge of the game – it was a fair result. But, we’re very proud of ourstart. We’d have accepted being second and two points off top of the league after four games.”
Harrogate Town: Oxley; Burrell (Fallowfield 81), McArdle, Hall, Page; Pattison, Falkingham, Kerry (Power 59); Thomson, Armstrong, Muldoon. Unused substitutes: Cracknell, Smith, Martin, Orsi.
Exeter City: Dawson; Sweeney, Ray, Hartridge; Key, Kite, Dieng (Edwards 65), Caprice; Jay, Brown, Amond (Nombe 65). Unused substitutes: Lee, Seymour, Rowe, Grounds, Dodd.
Referee: S Stockbridge (Tyne & Wear).