Brian Laws took charge of the champions for the first time with a trophy on the line but Rovers' disappointing season continued nonetheless in front of their own fans
Declan 'Fabio' O'Brien opened the scoring after the half-hour mark despite early pressure from the champions and a Kerrea Gilbert own goal put the Louth side two to the good before the break.
Hope was temporarily restored for Rovers through Killian Brennan's excellent free kick from the edge of the box before an Eric Foley shot ended their shot at the trophy minutes later.
Conor Powell saw a late red card for Shamrock Rovers as their disappointing season continued despite a new face in the dugout.
Three Airtricity Premier Division victories over Drogheda counted for nothing as Mick Cooke's side claimed the first piece of domestic silverware of the season.
Rovers came close to the opener 15 minutes in when a Gary McCabe free kick was deflected over the Drogheda crossbar for a corner. Minutes later and Gabriel Sava was tested through a Brennan strike.
Drogheda stemmed the pressure as Gavin Brennan's header tested Oscar Jansson but it wasn't long until Rovers were back on top.
Ciaran Kilduff fed Ronan Finn in the area but Sava was again on hand to make a smart save.
Drogheda then took the lead after riding out the early Shamrock Rovers pressure and gradually gaining a foothold in the game. A Gavin Brennan cross found Declan 'Fabio' O'Brien on the front post and he managed to edge the ball past the goalkeeper.
From there, Drogs took charge with Peter Hynes and Gavin Brennan forcing the best from Jansson and Paul Crowley heading narrowly wide.
Just before half time and Mick Cooke's side had a second. Phillip Hand's corner prompted an own goal from Kerrea Gilbert.
O'Brien could have added a third for Drogs early in the second half but his header was off target. That came before Rovers' false hope of Killian Brennan's strike on the hour mark.
The wind was taken from the Hoops sails moments later as Eric Foley's deflected effort restored Drogheda's two-goal cushion.
Rovers' misery was compounded 18 minutes from time when Conor Powell was dismissed for a foul on O'Brien as he raced clear onto the Rovers' goal.
Despite Rovers' late pressure it was Drogheda who came closest to adding to the scoring but Ryan Brennan's header couldn't beat Jansson.