Wexford Youths set their stall out early on as twice Paul Murphy found Danny Furlong who almost scored — the second effort needed Luke Gallagher to clear off the line.
Drogheda were slow to start but they opened Youths up on 26 minutes and Graham Doyle produced a fine double save to deny first Richie Purdy and then Adam Wixted.
Drogheda then had a chance of their own cleared off the line when Gareth McCaffrey looped a header over the onrushing Doyle, but Craig Delaney was on hand to clear the danger.
A minute into injury time and the hosts forced a corner which Griffin played short to Sean Brennan.
The Drogheda captain curled a fantastic shot into the top corner.
The hosts began the second in a similar fashion, Luke Gallagher struck a goal-bound shot from 40-yards just before Griffin was played through. But on both occasions Doyle proved superior.
The pressure finally paid dividends when, on 58 minutes, Griffin delivered a corner from the right which Kevin Farragher headed home from 12-yards out.
The game really opened up as Wexford began to threaten from counter-attacks with Gary Delaney forcing a save from McGuinness.
With ten minutes remaining though, the hosts were awarded a penalty as Marc Griffin appeared to be fouled.
Drogheda native Sean Thornton stepped up and scored the goal which ensured Drogheda’s swift return to the Premier Division.
There was late drama though as Youths had the ball in the net but the referee was overruled by his linesman and eventually awarded a free out for a bodycheck on McGuinness.
McGuinness, Deasy, Gallagher, Buckley (Farragher 20), Dunne, Hyland, Brennan (Thornton 54), Purdy McCaffrey, Griffin, Wixted (Ashe 90)
Dunican, Doyle, McKeown, Donnelly.
Doyle, McCabe, Delaney, Grace, Furlong, Keenan, Bonner, Whittle (Dempsey 86), Murphy, O’Connor, Kenny (Dunne 86)
Friel, Porter, Quinlan, George, Chambers
Graham Kelly (Cork)