Although Rovers led through Dan Carr, Pat Hoban levelled before Dan Cleary and Dane Massey belted in two goals around the hour to make it 3-1.
Rovers battled back through a Roberto Lopes finish but a double from Robbie Benson killed off the Hoops.
The lucrative Champions League qualification spot that Dundalk relinquished to Cork City after three years is back in their hands as the campaign nudges into the second half.
It had all started so well for the Hoops, edging ahead after just three minutes.
Carr, taking the mantle as their lead attacker in the absence of Graham Burke on international duty, stabbed home from close range.
The bizarre goal stemmed from Gary Rogers preventing Brian Gartland scoring an own-goal from Ronan Finn’s cross but Carr followed on the rebound.
Rogers, who like Burke had been part of the Ireland squad before, atoned for the misunderstanding on 12 minutes by denying Carr a second from close range and his loss through injury shortly after affected the hosts.
Dundalk took their time in settling but their deadly partnership of Michael Duffy and Hoban combined for the equaliser seven minutes before the break.
When Duffy regained possession on the left of the box, his pinpoint cross was nodded home by Hoban for the league’s top scorer’s 15th of the campaign.
Rovers then lost Ronan Finn to injury at the break and the floodgates opened. Firstly, on the hour mark, Dan Cleary made it 2-1 with a deft header from Robbie Benson’s cross.
It was particularly sweet for the Dubliner to score as he had trained with Rovers upon his release from Birmingham City without securing a deal. His gesture of cupping his ear towards Bradley was pointed, if not impetuous. He didn’t seem to mind accepting the mandatory booking.
Four minutes later and disaster struck for the Hoops with another concession. Dane Massey’s ability to rampage from left back is well known and his dribble towards the edge of the box and subsequent shot beat Tomer Chencinski at his near post.
Although Lopes tapped in from close range with 20 minutes left to pull one back, Benson hared away almost immediately to make it 4-2.
With Rovers chasing the game and the fans on their back, spaces opened up and there’s no better team than Stephen Kenny’s to exploit gifts.
Benson had been the talisman in the second half assault and he was once again offered too much space in the box to be able to turn and hammer in his second.
That was the cue for Rovers fans to turn on their young boss. The pressure has been coming for a while and as they remain midtable and in danger of losing their European spot the axe may well fall.
Chencinski; Boyle, Grace, Lopes, L Byrne; G Bolger, A Bolger (B Kavanagh 70); Coustrain, Finn (McAllister 46), S Kavanagh; Carr (Shaw 24).
Rogers; Hoare, Cleary, Gartland, Massey; Benson, Shields; Connolly (Adorjan 67), McGrath (Chvedukas 87), Duffy – Hoban (Tagbajumi 87).
Ben Connolly (Dublin)