The visitors made a strong start and Stephen Trimble in the Drogheda goal was a busy man as he made fine saves from Paul Corry and Graham Rusk in the opening quarter-hour.
Indeed, Rusk could have had a hat-trick as he headed across goal on another occasion, before drilling wide from eight yards.
The Students — and Robbie Benson in particular — seemed to be adopting a shoot-on-sight policy in the greasy conditions, but Drogheda survived that difficult spell.
Their best effort was a 30-yard piledriver from Hand which was parried by Ger Barron, and Stephen Quigley almost scrambled the ball in from the rebound.
Mulvenna also had two half-chances before he eventually broke the deadlock four minutes from the break. The move was instigated by a superb one-two involving the O’Brien brothers, and Colin’s cross from the right was turned in by the former Dundalk striker.
UCD replaced Darren Meenan with Samir Belhout at half time and the substitute made an immediate impression, breaking onto a Corry pass before forcing a fine diving save from Trimble.
The Students eventually levelled on 57 minutes when a Marshall corner was headed back across goal and Rusk nodded past Trimble from close range.
But Drogheda immediately restored their advantage when Dinny Corcoran’s pass released Mulvenna and he out-paced the two opposing centre-halves before shooting decisively past Barron.
Chances were at a premium after that, but UCD were given hope when Mark O’Brien earned two yellow cards in quick succession and was dismissed.
However, Drogheda sealed all three points in stoppage time when Colin O’Brien broke from halfway and fired past Barron from a tight angle.
DROGHEDA UNITED: Trimble, C O’Brien, McNally, Ekanem, Hand; M O’Brien, Gannon, Quigley (McCrossan 85), Freeman; Mulvenna (Nolan 80), Corcoran.
UCD: Barron; Harding, Leahy, D O’Connor, Nangle; Marshall, Corry, P O’Connor, Meenan (Belhout 46); Benson, Rusk.
Referee: Pádraig Sutton (Clare).