While it was Drogheda boss Mick Cooke's first trophy as a manager, the defeat marked an inauspicious first game in charge for Brian Laws, Rovers' "Director of Football", who saw his side finish with ten men.
Rovers were boosted before kick off as central defenders Craig Sives and Colin Hawkins passed fitness tests on respective ankle injuries to start in place of suspended pair Ken Oman and Pat Sullivan.
Conor Powell came in for the cup-tied Shane O'Connor at left-back in Rovers' only other change from their FAI Ford Cup defeat at Shelbourne.
Skipper Paul Crowley returned for Drogheda in their only change from their FAI Ford Cup exit in the quarter-final replay with to St. Patrick's Athletic last Monday.
In a slow burner of a game, we'd to wait 16 minutes for an effort on goal and that, a Killian Brennan shot from distance, scarcely troubled Drogheda keeper Gabriel Sava.
Three minutes later Rovers' keeper Oscar Jansson had to move a little more sharply to get down to hold a diving header from Killian's brother Gavin from Eric Foley's superb diagonal ball.
But the game should have had the goal it craved to spark it to life on 21 minutes.
Hoops' winger Gary McCabe and striker Ciaran Kilduff combined well to put Ronan Finn through one-on-one with Sava.
But Finn showed an uncharacteristic lack of composure as he drove the ball low and hard with Sava saving with his feet.
It was a bad miss Rovers would rue as Drogheda took the lead on 32 minutes.
Gavin Brennan worked a short corner with Foley to whip a low cross in from the end line where Declan "Fabio" O'Brien showed his clinical striker's instinct to tuck the ball home at the near post.
It would get worse for Rovers as Drogheda twice came close before doubling their lead by half-time.
On 38 minutes Jansson made a one-handed save at full stretch to touch away a header by Gavin Brennan from Brian Gannon's cross.
From Foley's resultant corner, skipper Paul Crowley powered a header narrowly.
Rovers looked rattled and a second goal did arrive two minutes before the break when Philip Hand's low corner was punched by Jansson against right-back Kerrea Gilbert with the ball ricocheting to the net for an own goal.
Rovers looked to have got right back into the game on 59 minutes when Brennan found the top corner with a brilliant free-kick.
But Drogheda hit back within two minutes. A half clearance from Colin Hawkins ran kindly for the in-rushing Foley whose drive deflected off Hoops' skipper Stephen Rice to wrong-foot Jansson and hit the back of the net.
Powell being shown a straight red for pulling down O'Brien in a last-man challenge on 72 minutes just about summed up Rovers' day.
SHAMROCK ROVERS: Jansson; Gilbert, Sives, Hawkins, Powell; Finn, Rice, Brennan, McCabe (Dennehy, 60 mins.); Kilduff (Twigg, 65 mins.) , Stewart (Kavanagh, 60 mins.).
DROGHEDA UNITED: Sava; Gannon, McNally, Prendergast, Hand; Foley, Crowley; O'Brien (S. Brennan, 90+2, mins.), R. Brennan, G. Brennan; Hynes (Mulvenna, 90 mins.).
Referee: Tomas Connolly (Dublin)
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