Jack Byrne sent Stephen Kenny a timely reminder of his class as he inspired the champions to the top of the Premier Division.
The Ireland midfielder delivered a first half masterclass at Tallaght Stadium, creating the opening goal for Sean Gannon and then adding the second before the interval.
Stephen Bradley caused some nervous moments for The Hoops with his 57th minute replacement of Byrne even if substitute Graham Burke added the insurance of a third goal nine minutes from full time.
For United clawed a goal back through Evan Weir in the 64th minute and took advantage of Byrne's absence to take control for a period midway through the second period.
Ultimately, The Hoops had too much power, guile and quality with the late introduction of Richie Towell, Ronan Finn, Rory Gaffney and Aidomo Emakhu proving both their vast resources and their ability to turn a match within a few minuets.
Following Friday's defeat in Derry City, this was the perfect pick me up ahead of Friday's clash with St Pats and Friday week's date with Bohemians.
United were sprightly in the opening minutes and made some headway down their left.
But Darragh Markey skied his second minute shot while Alan Mannus was alert three minutes later to parry Adam Foley's fiercely struck effort from a tight angle away from the danger area.
Byrne was finding himself on a different wavelength during this spell with his passing but once he began to get into the groove, he sliced open the United defence with an ease that will have worried Kevin Doherty in the visitors' dugout.
Two corners in the space of a minute gave Hoare a half chance with Lopes heading the second delivery inches over the crossbar.
The Boynesiders' rearguard was struggling badly from set pieces and another Byrne corner resulted in Lopes heading just wide in the 18th minute.
An opening goal appeared to be a formality and so it proved in the 21st minute although it still required a moment of magic from Byrne to create the vision and the pass for the overlapping Gannon to beat McCabe with a low drive from just inside the box.
With their early nerves now a distant memory and Byrne lording it in the middle of the park, Rovers took a vice-like grip on the proceedings.
The only surprise was that it took them until the 42nd minute to double their advantage.
This time, Byrne opted to try his luck from just outside the United penalty area with his effort possessing both accuracy and pace to find the top corner of Colin McCabe's net.
The Hoops resumed the second period in the same vein with Byrne's free kick finding Gannon's head but his effort was comfortably saved by McCabe.
Bradley then opted to replace Byrne with Graham Burke in the 57th minute, obviously with a view to his side's hectic March that plays out with games against St Patrick's Athletic on Friday, Bohemians on Monday and then Dundalk and SLigo before the international break.
Understandable given his side's dominance at this point and their demanding schedule.
But making sure of these three points were the immediate priority and when Evan Weir scored Drogheda's first goal of the season with a close range finish after Mannus parried Roughan's cross, the decision looked somewhat dubious
Rovers were hardly under the cosh but their earlier superiority had been eroded by United's persistence.
However, as the game entered its final 20 minutes the home side once again asserted control, helped by the triple arrival into the game of Ronan Finn, Rory Gaffney and Aidomo Emakhu in the 76th minute.
Two minutes later Richie Towell replaced Dylan Watts and nine minutes from the end, United's cheap concession of the ball in the Rovers' half as they pressed for an equaliser allwoed Finn to feed Burke who ran 50 yards to beat McCabe with a well taken low shot.
Shamrock Rovers: Mannus; Hoare, Lopes, Grace;Gannon (Finn 76) ,Watts (Towell 78), O'Neill, Lyons; Byrne (Burke 57), Greene (Gaffney 76) , Mandroiu (Emakhu 76). Subs- Pohls, Leitis, Cotter, Carey,.
Drogheda Utd: McCabe; Poynton, Quinn, Roughan (Lyons 79), Massey, Weir; Markey, Deegan, Clarke, Grimes (Hughes 59); Foley(Williams 59). Subs -Davis, Boudiaf, Rooney, Mutawe, Supple, Haddock.
Referee: Adriano Reale (Clare). att 4,107.
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