Among the bookies' favourites for relegation from League Two this term, Simon Weaver's Sulphurites took their much-fancied opponents apart, running out victorious thanks to goals from Alex Pattison, Matty Daly and Jack Muldoon.
The opening 20 minutes were shaded by the hosts, who played with good intensity and moved the ball neatly through the thirds in their new 3-4-2-1 formation.
Clear-cut chances were at a premium, though Pattison ought to have been awarded a spot-kick when he burst between two defenders inside the away box and went down under contact from Remeao Hutton.
The Robins began to grow into the contest around the midway point of the half, though it was Harrogate who created the first decent opening of the afternoon on 42 minutes.
Daly and Luke Armstrong combined just outside the Swindon area, the latter's clever pass freeing up Pattison to get off a low effort which Soloman Brynn got down well to push away.
The resulting corner from the left was initially nodded clear from inside the Swindon six-yard-box, but kept alive by the head of Miles Welch-Hayes before landing for Pattison, who struck it on the bounce and lashed an effort home via the underside of the cross-bar.
Town began the second period on the front foot, with Pattison twice able to drive deep into Swindon territory before being crowded out.
They then got away with one as debutant goalkeeper Pete Jameson came for Louis Reed's driven left-wing corner and didn't get there, Mathieu Baudry unable to direct his back-post header inside the upright.
That scare survived, the hosts went about doubling their advantage with 51 minutes on the clock.
Pattison and Armstrong combined on the edge of the box, and although the Robins' defence managed to prevent either man from getting in behind, the ball was eventually worked into the path of Daly who smashed an unstoppable effort into the top corner.
Harrogate looked comfortable at 2-0, however Swindon were handed a lifeline in the 65th minute when Miles Welch-Hayes caught Jacob Wakeling as he attempted to clear the ball and Tom Reeves pointed to the spot.
But, Ben Gladwin's low spot-kick hit the foot of the left-hand post on its way wide of the goal.
Town took full advantage of that little bit of fortune, wrapping up a fully-deserved opening-day three-point haul in the 76th minute.
Jaheim Headley and Josh Austerfield initially showed real composure in calmly playing the ball out from the left-back position before the former found Daly near halfway.
The former Everton man then slid a nicely-weighted pass in behind for substitute Muldoon, who got the better of the advancing Brynn on the edge of the away area and was able to roll into an empty net.
The Robins did muster a couple of late chances to pull one back, Harry McKirdy shooting tamely at Jameson when well placed before Cian Harries only managed to clear the cross-bar when left free inside the Harrogate box.
Town also came close to adding a fourth themselves after excellent play down the left by Headley teed up Armstrong for a strike which Brynn saved well.
They had however already done more than enough to seal a stunning result, one which sees them finish the opening weekend of 2022/23 second in the League Two standings.