With 76 minutes on the clock and the Sulphurites leading 3-2, Brahima Diarra was wiped out by Jake Ball deep in home territory, but the man in the middle initially allowed play to continue.
Jack Muldoon picked up the ball and played a pass to Luke Armstrong on the edge of the penalty area, leaving Town's leading marksman with a clear run on Joel Coleman's goal.
But, at that moment, Mr Hair opted to blow his whistle, denying Harrogate what looked a clear advantage by awarding a free-kick and showing Ball a yellow card.
The visiting players and management team were visibly angered by the decision at the time, and Weaver subsequently spoke to the referee after the full-time whistle.
"It's a really bad error from the referee, isn't it. He's a nice guy, as pleasant a referee as you will meet, but today he made a bad mistake," the Town boss said.
"He held his hands up and said to me 'I know what you are saying, I got it wrong'. He's admitted it, but that's probably cost us.
"He said to me 'well, your defenders made some mistakes as well' and yes, he's right, but it's frustrating because we were working so hard.
"That's the difference in levels of referee. I've always said that they can give themselves a couple of seconds just to see if there is an advantage to be given.
"I think that separates [referees]. It's like time on the ball for players, the better ones seem to have a bit more and it's the same between the levels of referees."
Mr Hair’s decision-making may have disappointed Weaver on that occasion, but the Cambridgeshire official didn't get much wrong on the day, awarding Town a 24th-minute penalty, as well as turning down two hopeful Rochdale appeals for spot-kicks in the second period.
And Alex Pattison converted that penalty to put the visitors 2-1 up after Max Clark had cancelled out his stunning early opener.
Pattison then released Jack Diamond down the left wing to fire home and put Harrogate in full control of proceedings shortly after half-time.
Dale substitute Abraham Odoh’s close-range finish reduced the deficit before Armstrong was denied the opportunity to put the Sulphurites 4-2 up by Mr Hair’s unwillingness to play the advantage.
And that moment was to prove pivotal as Tahvon Campbell struck in the 87th minute to earn the hosts a share of the spoils.
Saturday’s result means Town remain 13th in the League Two standings with 38 points to their name from 29 games.