The game was finely poised at 1-1 midway through the first period when the Boynesiders had Richie Purdy sent off for a late tackle, and after City scored again either side of half-time it was damage limitation for Pete Mahon’s team.
Whether they would have got anything out of this game with 11 men on the field is a moot point, though, as the Leesiders were the better team throughout.
The Leesiders lined out with two changes from the team who convincingly defeated Galway last week, with Garry Buckley and Karl Sheppard replacing Jimmy Keohane and Stephen Dooley, while Drogheda boss Pete Mahon named the same starting XI that put St Pat’s to the sword.
Straight from the first whistle, it was Cork who took a firm grip of the game, and last season’s league top Sean Maguire passed up a great scoring chance inside three minutes.
Drogheda’s Ciaran McGuigan was booked for a mistimed challenge out on Cork’s right flank and Greg Bolger rolled the free kick across the 18-yard line to Maguire, who completely miskicked when he had time to pick his spot.
That was an escape for Drogheda, but they only remained on level terms until eight minutes in, when a corner kick was only partially cleared before being headed back towards goal and Maguire beat Drogs keeper Stephen McGuinness with a firm header from six yards.
Drogheda could easily have conceded again quickly as Sheppard had a shot blocked by the feet of Stephen McGuinness and Steven Beattie fired over the bar.
But the home side survived and hauled themselves back level on 17 minutes. Jake Hyland teed up Stephen Elliott, whose mishit effort was scrambled round the post by Cork keeper Mark McNulty.
Thomas Byrne swept the corner to the edge of the penalty area and McGuigan fired the ball back into the six-yard area, where Elliott’s back-flick nestled in the Cork net.
Drogheda’s celebrations had barely subsided, though, when they were reduced to men, with Purdy dismissed on a straight red card for a poor sliding challenge on Beattie.
That left the Boynesiders with a mountain to climb, but when Maguire blazed narrowly wide from 18 yards, with McGuinness well beaten, it seemed that the 10 men might hold out until half-time.
Cork were clinical the next time they visited the home penalty box, though, with Gearoid Morrissey sticking out a boot at the near post to deflect Beattie’s corner kick past McGuinness for the lead.
For all that, in first-half stoppage time the Drogs nearly grabbed an unlikely equaliser as Hyland swivelled in the box and fired narrowly over McNulty’s crossbar.
Cork heeded that warning and, within five minutes of the restart, doubled their advantage from another corner. Beattie’s cross was punched by McGuinness towards Sheppard, who fired straight to the net through a crowd of players from the edge of the box.
Sheppard was put through on goal three minutes later and this time McGuinness deflected the ball wide.
Cork boss John Caulfield replaced Maguire, who came into the game an injury doubt, with Achille Campion short of the hour mark, and the Frenchman should have put City out of sight shortly after coming on after being set up by the excellent Morrissey, but after stealing a yard on his marker he swept the ball narrowly wide of the upright.
The Leesiders were cruising at this stage and Drogheda were indebted to their keeper McGuinness again on 78 minutes when he saved well at the feet of substitute Conor Ellis.
But Ellis eventually got on the scoresheet deep into stoppage time to complete a convincing win.
McGuinness, Deasy, Gallagher, McGuigan, Dunne, Purdy, Thornton, Hyland, Byrne, Elliott, G Brennan.
Elworthy for Byrne (55min), Buckley for Gallagher (62min), McCaffrey for Elliott (71min).
McNulty, McCormack, Bennett, Delaney, O’Connor, Beattie, Morrissey, Bolger, Buckley, Sheppard, Maguire.
Campion for Maguire (56min), Ellis for Morrissey (75min), Ellis for Morrissey (75min).
Derek Tomney (Dublin).