The hosts, whose poor recent form has seen them plummet to second-bottom in the Airtricity Premier Division, started brightly with Keith Buckley forcing a fine save from Gabriel Sava in the third minute following a jinking run by Dwayne Wilson.
However, the defensive uncertainties that have undermined their campaign to date resurfaced two minutes later as their failure to deal adequately with a Jason Marks cross resulted in Stephen Quigley powering a rising, left-foot shot beyond Dean Delany from 16 yards.
That lead could well have been doubled in the 11th minute as an intricate passing move down the right wing culminated in Marks’ gaol-bound attempt blocked to safety by Ryan McEvoy.
The enterprising opening was maintained with Sava almost gifting Bohemians an equaliser in the 18th minute as he misjudged a McEvoy free-kick, but his blushes were spared as the ball rebounded back into his grasp via his right-hand post.
Within five minutes, the hosts almost restored parity when Chris Lyons’ driving run culminated in Buckley’s low shot from the edge of the penalty area beating Sava but Paul O’Conor was perfectly positioned to clear off his own line.
The hosts look well set at this stage to claw their way back into contention but their deficit was doubled in the 38th minute as Marks cut inside Michael Barker far too easily and his resultant cross was despatched emphatically to the net by the head of Declan O’Brien.
O’Brien should have settled the tie a minute before the break as he raced clear of a ponderous home defence but his lob over Delany lacked the precision of his earlier attempt and sailed harmlessly over the crossbar.
Although the impressive Buckley threatened with a curling free-kick within a minute of the restart, further defensive uncertainty allowed O’Brien to seize possession in the 51st minute but his audacious chip narrowly cleared the crossbar of a back-pedalling Delany.
The hosts continued to create chances at the opposite end and the clearest of the evening arrived three minutes later as substitute Kevin O’Leary showed great pace in outstripping Alan McNally but Sava was quickly off his line when denying him from 14 yards.
Another substitute was to effectively confirm United’s progression as a busy Cathal Brady teed up Philip Hand, whose swerving shot from 20 yards deceived Delany en route to the net via the crossbar.
The contest petered out from that juncture with the visitors comfortably repelling the progressively sporadic bouts of home pressure, ensuring their emphatic advancement to the third round.
BOHEMIANS: Delany; Barker, Lopes, Mulcahy, Byrne; Wilson (O’Leary, 46), Buckley, McEvoy, Pender, Mulligan (Moore, 75); Lyons (Kelly, 70).
DROGHEDA UNITED: Sava; Quigley, McNally, Byrne, Grimes; Brady, O’Conor, Brennan (Daly, 86), Marks (Hand, 58); Hynes, O’Brien (Cassidy, 64).
Referee: Neil Lyons (Dublin).