Three defeats in four matches had seen the Sulphurites drop from second in the League Two standings and down to seventh place prior to Saturday’s 2-1 comeback victory
And although his side did not earn themselves any league points as they secured a first win in five attempts, Weaver believes the “stirring” nature of their late recovery is something that can be built upon.
“We want it to be a turning point. We want to turn the ship again and make sure we have momentum and wind behind us,” he said.
“There’s nothing quite like the FA Cup and real stirring victory like we had today to get that feel-good factor back and get people buoyant again because we are in a results-based business.
“While it was hard work and a war of attrition for 70 minutes, the last 20 have excited us all. We played at a pace that I don’t think the opposition could handle.
“The character shone through in the end and that is why we won the game. We’ll learn from this today and it’s a better feeling having won the game and everyone being positive.
“We conceded one goal, but the defending the last 10 minutes was resolute, it was gritty and we found a way to win.”
Jordan Ponticelli's close-range finish handed Wrexham a first-half lead against the run of play at Wetherby Road, and with Town looking badly out of sorts once again, it appeared as if that goal would be sufficient to seal the Welsh outfit's progress into the second round.
But, a change in formation and the introduction of players off the bench sparked the hosts into life. They levelled on 73 minutes when Jack Diamond touched a free-kick into the path of Simon Power, who drilled past Robert Lainton from 19 yards out.
Six minutes later, the turnaround was complete, Danilo Orsi slotting into an empty net after Lainton had been forced to dash from his goal-line to deny Jack Muldoon one-on-one.
Harrogate Town: Oxley, Sheron, Burrell, Hall, Page (Fallowfield 46); Pattison, Falkingham, Kerry (Orsi 62); Thomson (Power 57), Martin (Muldoon 62), Diamond. Unused substitutes: Cracknell, Smith, Wilson, Williams.
Wrexham: Lainton; French, Hayden, Tozer, Brisley (Angus 82), Green (Jarvis 83); J Jones (Hall-Johnson 83), Davies, Young, McAlinden (Hosannah 69); Ponticelli. Unused substitutes: Dibble, Reckord, D Jones, Lennon, Bickerstaff.
Referee: S Oldham (Lancashire).