The Hoops suffered no fatigue after their heroics against Tottenham in the Europa League at White Hart Lane on Thursday.
Manager Michael O’Neill and his players, with a game in hand, are obviously intent on retaining the title.
Rohan Ricketts set up an early chance for Rovers with a clever ball through to Karl Sheppard, only for the striker to see his effort brilliantly pushed round the upright by
Rovers were in front on 23 minutes when an inswinging corner by Ricketts on the left was headed home from close range by Sheppard.
Ciaran Kilduff failed to get the final touch when a Sheppard cross unhinged the Drogheda defence. Then Conor McMcCormack had a 14-yard effort effort deflected for a corner by Hand.
All that pressure was bound to tell and the Hoops stretched their lead on 41 minutes when Sheppard played a probing ball through to Kilduff who cleverly rounded the keeper before planting the ball in the unguarded net.
Little was seen of Drogheda as an attacking force although substitute Stephen Quigley did have one long-range straight at keeper Richard Brush early in the second half. Turner and Ricketts both went close as the Hoops continued to dominate.
In one breakaway attack for the visitors Peter McMahon had one good effort well saved by the diving Brush after Gavin Brennan had set up the chance with a cross.
Rovers eventually stretched their lead on 70 minutes when Brian Gannon was yellow carded for taking down Kilduff inside the box. The resulting penalty was hammered into the roof of the net by substitute Billy Dennehy.
The Hoops had a fourth goal on 81 minutes, this time with Dennehy the creator, to allow substitute Gary O’Neill to fire past Trimble from 12 yards.
SHAMROCK ROVERS: Brush; Sullivan, Sives (Dennehy 45), Murray, Stevens; Rickett’s, McCormack (Finn 77), Turner, Paterson; Kilduff, Sheppard (O’Neill 71).
DROGHEDA UNITED: Trimble; Gaul, McNally, Rogers, Hand; Fitzgerald (Quigley 45), Gannon (Mulvanna 78), McMahon, Brennan; White (Corcoran 59), Lynch.
Referee: Tom Connolly (Wicklow).
On Saturday, Sligo cruised to a 7-1 win against 10-man Galway United at the Showgrounds.
Rovers hit four goals in the opening half. On 15 minutes Aaron Greene fired home and John Dillon smashed home the second from the penalty spot after Ronan Caldwell was red-carded for a foul on Greene.
Matthew Blinkhorn made it three before and Alan Keane smashed home the fourth before the break. Blinkhorn’s second followed early in the second half, before Oscar Sibanda pulled one back for Galway.
But Blinkhorn’s third and a seventh goal from Raffaele Cretaro at the death eased Rovers to victory.
In the First Division 10-man Longford beat Waterford 2-1.
Thomas Crawley put Longford ahead and Graham Dowling doubled the lead. Gary Dunphy pulled one back for Waterford.
Longford held on despite losing Northern Ireland and one-time Manchester United winger Keith Gillespie sent off at the death.
Meanwhile, just over two weeks after Limerick staged a late, late show to beat Shelbourne and open up the First Division title and promotion race, the two clubs will meet again at Tolka Park this evening in the FAI Ford Cup quarter-final. Kick-off is at 8pm.
And, in the Airtricity League Premier Division, UCD host St Patrick’s Athletic at the UCD Bowl, kick off 7.45pm.