The home side enjoyed the lion’s share of possession with several good chances falling to both Kevin McHugh and Adam Clarke.
However despite enjoying a great first half hour, it was the visitors who took the lead on 42 minutes.
Tom Elmes skipped into the box from a throw-in and drilled home from close range to make it 1-0.
The second half was a much more open affair with both sides going close.
Thomas McBride had a great chance for Harps but Wexford keeper Doyle did well to parry the effort over the crossbar.
Wexford has shouts for a penalty turned down but they did get a chance from the spot in the 90th minute when Shane Nolan was taken down but Wexford squandered the chance.
Harps will rue many missed chances but Felix Healy knows that uncertainty in front of goal is costing him points and results.