A goal in either half saw Welling's winless run in National League extend to 18 games on Tuesday night.
Welling were never at the races against a side under the interim management of former Wings boss Neil Smith.
The Wings have now gone 450 minutes without a goal and remain rooted in the relegation zone.
Welling faced an uphill task as early as the ninth minute. Poor defending allowed Lee Minshull to twice win the ball. He fed Rohdell Gordon, who went past Noah Chesmain with ease and fired in via the underside of the crossbar.
As an attacking threat, Welling lacked any quality in the final third. Defensively they were rarely stretched but goals have been their problem this season and Dean Frost is no closer to solving it than former boss Loui Fazakerley.
Before half-time, Tomi Adeloye should have opened his Welling account but his low shot was kept out by the legs of keeper Alan Julian.
As has been the norm for most of the season, Welling started the second half slowly.
An error by Nortei Nortey let in Minshull but his shot was parried away by keeper Marek Rodak, then the lively Moses Emmanuel sent his effort crashing back off the crossbar.
Welling appeared to have weathered the storm but they conceded a needless penalty on 55 minutes. Ian Gayle held onto Emmanuel in the penalty area and Bromley's top scorer then drilled home the spot-kick.
Welling had not scored in their previous four games so the points were well beyond them at this stage. Daniel sent a free-kick over the bar and Julian saved from the same player but that was as good as it got for the visitors.
Welling's evening was summed up with the last kick of the game. Substitute Sahr Kabba set up Michael Bakare on a three on two break and he blazed over when he should have scored.
They host leaders Cheltenham on Saturday and face a near-impossible task to maintain their National League status.
Welling: Rodak, Williams, Gayle, Lokko, Chesmain, Daniel (Bakare 75mins), Nortey, Kargbo, Harris, Wanadio, Adeloye (Kabba 77mins). Subs not used: McEntegart, Ferry, Vidal. Attendance: 1,110.