Full back Alex Graham opened the scoring with a converted try to give the Tigers a 7-0 lead, but they found themselves behind before winger Jamie Broadley went over to level the scores at 14-all at the interval.
Broadley scored his second after the break to restore the visitors the lead which proved decisive, as captain Tom Outram kicked nine points.
Harrogate are bottom of the league as the Tigers move on to ten points from their first five games.
Sheffield are up to fourth place in National Three Midlands after a decisive 50-29 victory over Longton at Abbeydale Park.
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After conceding an early penalty, Sheffield hit back with two tries by No 8 Jonny France, both converted by Harry Pratt, only for the visitors to reply with two more tries to lead 17-14.
A third try by winger Rupert Broadley just before half time put the hosts 19-17 ahead, and they pulled clear after the break with two more tries by France, two by full-back Fergus Wood and an eighth by flanker Tom Womack.
Dinnington threw away a good chance to make progress in North One East when they went down 41-28 at Durham City.
Dinnington, still well below their potential with players missing, were leading 28-24 until lapses in concentration let Durham in for two converted tries and a penalty goal in the last ten minutes.
Their consolation was a four-try bonus, the touchdowns coming from stand-off Jonny West, flanker Michael Leitch, prop John Marshall and skipper and No 8 Ryan Donnelly. West converted one and landed a penalty and drop goal.
Wath on Dearne suffered a 36-26 defeat at Ripon in Yorkshire Two despite a second half revival. Wath had two first-half penalty goals by fly-half Jack Whitlam and tries by Ben Cairns, Andy Hopkin and Matt Burrows, one converted by Whitlam.
Barnsley, also playing in Yorkshire Two, moved up with a 26-14 win at Sheffield Medicals.
Full-back Callum Emery, who has taken over the goal kicking from Andy Appleyard, landed two conversions and four penalties as well as racing over for one of the visitors’ two tries.
Previously without a win, Stocksbridge lifted themselves off the bottom of Yorkshire Three with a 17-10 home win over Halifax Vandals.
Rotherham Phoenix ran in seven tries in a 43-8 win at Stanley Rodillians in Yorkshire Four, but Rotherham Clifton went down 27-15 at home to Bramley Phoenix.