After dominating the early exchanges, the hosts came closest to finding the breakthrough when Tyler Denton's cross found Marcus Antonsson, who in turn saw his looping header bounce off the crossbar.
The Championship side continued to press forward and came close again through Souleymane Doukara, the powerful forward shaking off a challenge from Josh Falkingham before forcing goalkeeper James Belshaw into a smart stop.
Despite some resilient defending from Town, Leeds were awarded a penalty in the 15th minute when Ben O'Hanlon went to ground on Doukara and handled in the box, with Kemar Roofe dispatching the resulting spot-kick to break the deadlock.
Leeds continued to dominate possession and almost grabbed a second when Antonsson found space in the Town area but the Swede saw his effort clawed away by the impressive Belshaw.
The hosts doubled their lead just after the half-hour mark through Madger Gomes, the youngster latching onto a loose ball on the edge of the Town area to find the bottom corner with aplomb.
In a half dominated by their hosts, Harrogate struggled to get a foothold in the game and went in two goals down at the break.
Both Weaver and Thomas Christiansen made eleven changes to their teams at half-time as 22 new players took the field for the second-half, with Town introducing last season's talisman Simon Ainge and new signing Terry Kennedy.
Leeds began the second-half as they ended the first and found a third shortly after the restart through midfielder Pablo Hernandez. Luke Ayling's deep cross found Stuart Dallas who teed up the Spaniard to fire the Whites three in front.
With 15 minutes to play, Jordan Thewlis lead a Town breakaway and outpaced the Leeds defence before beating substitute goalkeeper Felix Wiedwald to pull one back.
Town continued to press and were rewarded again in the 83rd minute when Ben Pollock nodded home a deep cross to set-up an interesting final few minutes at Thorp Arch.
However, the Championship side showed their quality and restored their two-goal advantage just two minutes later through Lee Erwin, the striker poking in from close range to mark the end of a fascinating six-goal thriller that left both sides with plenty of positives.
Harrogate first half: Belshaw, Fallowfield, Middleton, Lees, O'Hanlon, Thomson, Falkingham, Agnew, Leesley, McIntosh, Knowles.
Harrogate second half: Hallwood, Gooda, Kennedy, Pollock, Parker, Day, Burrell, Emmett, Trialist, Ainge, Thewlis.
Leeds first half: Green, Denton, Cooper, Diagouraga, O'Connor, Gomes, Bridcutt, Klich, Roofe, Doukara, Antonsson.
Leeds second half: Wiedwald, Berardi, P. McKay, Bellusci, Ayling, Dallas, Phillips, Vieira, Sacko, Hernandez, Erwin.