The deadlock was broken after 10 minutes when Peter Hynes’s overhead kick was judged to have crossed the line.
Declan O’Brien added the second and Gavin Brennan a third before the break.
Referee Neil Doyle showed a straight red card to Mayfield sub Stephen Hurley for an off-the-ball challenge on Eric Foley in the 71st minute, before the 10 men pulled a goal back thanks to a Greg Kelly header.
Drogheda finished the stronger, however, with Breen striking in the 84th, 88th and 92nd minutes to underline his side’s dominance.
DROGHEDA UNITED: Sava; Gannon, McNally, Prendergast, Hand; Brady (Marshall 53), Foley, Crowley, G Brennan (S Brennan 46); Hynes (Breen 63), D O’Brien.
MAYFIELD UNITED: Winters; Kiely, Wolfe, I Fleming, Roddis; E Fleming (McBride 60), O’Shea, Grainger, Warren (Buckley 77), Kelly; Finn (Hurley 66).
Referee: Neil Doyle (Dublin).