The visitors were set up to contain but that plan had to be thrown out after six minutes when Chris Forrester beat Dave Ryan at his near post.
Christy Fagan added a second after diverting Ian Bermingham’s shanked effort into the net 10 minutes later.
Drogheda’s misery deepened on 40 minutes when Ryan allowed Conan Byrne’s cross from the right to drift in at his near post.
Hopes of a better showing in the second half from Drogheda faded after just three minutes when Forrester netted.
The fifth arrived in similar fashion when substitute Mark Quigley put Byrne in to drill low into the net.
Despite ending with 10 men after Killian Brennan had gone off injured, Byrne put the finishing touch on a fine team move to claim his hat-trick.
ST PATRICK’S ATHLETIC: B Clarke (R Gilmartin 45); C McCormack, S Hoare, K Browne, I Bermingham; K Fahey; J Chambers, K Brennan; C Byrne, C Fagan (M Quigley 62), C Forrester (D Kavanagh 71).
DROGHEDA UNITED: D Ryan; G Holohan, D Doyle, P Andrews, G Brennan; C Walshe, P Crowley, M Daly, S O’Connor, C Brady (E Foley 56); G O’Neill (P Hughes 56).
Referee: Dave McKeon (Dublin).