The visitors came into the game just two places above the drop zone, yet took an early lead and never looked in too much danger of losing it on a sunny afternoon at the CNG Stadium.
Simon Weaver’s team’s first-half display was flat and Wood looked much the better side, edging in front after just nine minutes.
A right-wing corner was taken short, then delivered into the box where it was all too easy for Tom Champion to glance a near-post header past James Belshaw.
Falling behind failed to shock Town into life, though they did eventually begin to string a few passes together.
They finally created an opening of note in the 25th minute when Liam Agnew's burst forward and neat pass sent Jack Muldoon in on goal, however he could only fire into the side-netting.
Callum Howe then met Joe Leesley's right-wing free-kick and nodded back across goal for Warren Burrell to stab an effort over the bar.
Despite being held by Champion, Howe got on the end of another Leesley set-piece after 43 minutes, forcing Ryan Huddart to tip his back-post header over the top.
Things then almost got worse for Weaver and his men shortly before the interval when the ball dropped for Ralston Gabriel just outside the six-yard box, though a superb covering tackle by Kelvin Langmead saved the day.
Another crucial last-ditch intervention by Harrogate's veteran centre-half was required just minutes after the resumption, Langmead throwing himself in front of a goal-bound Justin Shaibu strike and diverting the ball behind.
There was little forthcoming at the opposite end of the field until the 68th minute when Muldoon saw a shot blocked following a scramble in the away penalty area.
Substitute Jordan Thewlis then pulled back from the left for Liam Kitching, but his first-time strike was just about gathered by Huddart at his near post.
It took until the last five minutes for Town to really have a go at their visitors, substitute Aaron Williams spinning and firing tamely at Huddart before a series of penalty appeals in quick succession were all turned down by referee David Richardson.
It was Wood who looked the more likely in the dying moments, a lucky bounce setting Bradley Ash clean through on goal, however Belshaw did brilliantly to deny the visiting attacker as he attempted to dance around him.
The final whistle then arrived with the away side still pushing for a second, defeat meaning that Town have now won just one of their last six matches.
They remain sixth in the National League standings, but now sit only a single point ahead of seventh-placed Eastleigh with a difficult trip to top-of-the-table Leyton Orient up next.