College had an early chance when midfielder Paul O’Conor put Chris Lyons through only for the striker to awkwardly loft the ball over the bar.
Drogheda went close to breaking the deadlock on 12 minutes when Tiernan Mulvenna played a ball over the top of the UCD defence but Declan ‘Fabio’ O’Brien could not finish.
The visitors had another chance when Ryan Brennan supplied a perfect pass on the right to O’Brien. This time the striker’s angled shot was narrowly off target.
Drogheda again went close on 35 minutes when a Gavin Brennan cross from the left allowed his brother, Ryan, to guide a header narrowly over the top.
O’Brien had appeals for a penalty ruled out by Welsh referee Nick Pratt on 43 minutes after appearing to be taken down in the area by defender Danny Ledwith.
UCD then went close early in the second half when an Ashley Grimes tackle on Robbie Benson enabled the ball to fall nicely into the path of defender Danny Ledwith, only for his effort to end in the sidenetting.
O’Brien then had a header cleared off the line before Tiernan Mulvenna had a shot blocked on the line by an overworked home defence.
Perhaps against the run of play College took the lead on 69 minutes when Graham Rusk was dispossessed by the Drogheda defence but Ledwith was able to collect the ball and fire home.
Tiernan Mulvenna and Derek Prendergast both had efforts which rebounded off the woodwork as Drogheda frantically threw everything forward in search of an equaliser.
But they had their reward on 82 minutes when substitute Sean Brennan headed across goal for skipper McNally to nod home from close range.
UCD: McGinley; Douglas, D O’Connor, Leahy, Ledwith; Burke (Langtry 46), Corry, P O’Conor, Benson (Morrison 63); Rusk, Lyons (Belhout 75).
DROGHEDA UNITED: Sava; Quigley (Brady 75), Prendergast, McNally, Grimes Gannon, R Brennan (S Brennan 71), Foley, G Brennan; O’Brien, Mulvenna.
Referee: Nick Pratt (Wales).