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A stunning free-kick by top scorer Adam Coombes sent Welling into the last 16 of the FA Trophy on Saturday.
Coombes scored a brilliant 30-yard free-kick, similar to his goal at Dartford earlier this season, to win the game with seven minutes left.
It was Coombes' 30th goal of the season and gave Welling a place in Monday's third round draw.
Welling hit the woodwork inside four minutes. A misplaced pass by the hosts gave Christian Nanetti the chance to run at the home defence and his left-foot shot clipped the top of the crossbar.
Bromley took a sixth-minute lead when a free-kick by Joe Anderson fell perfectly for captain Jack Holland, who had the presence of mind to backheel the ball home with keeper Chris Lewington grounded.
Welling nearly equalised on 13 minutes but keeper Alan Julian turned over Coombes' long-range effort.
Bromley would have doubled their lead seven minutes later but Louis Dennis strayed offside after Welling keeper Chris Lewington came rushing out to clear but never got close to Blair Turgott.
Welling were on level terms on 31 minutes. Danny Waldren crossed the ball from the left wing and, with Harry Crawford failing to get a touch at full stretch, keeper Julian could only watch as the ball crept inside the far post.
Lewington produced an excellent save with his legs to deny Turgott after he beat the offside trap on 39 minutes, exposing a horribly square Welling defence.
Welling had the first chance of the second half as Crawford's shot was beaten away by Julian, the Wings striker then missing from close range from a corner.
It was the Wings who continued to press although they failed to create too many openings.
Bromley's Lee Minshull shot over from close range and Turgott wide from distance as the hosts created late chances.
But it was Welling who edged ahead thanks to Coombes' moment of brilliance.
Bromley: Julian, Wynter, Johnston, Holland, Minshull, Cunnington, Dennis (Martin 63mins), Anderson, Higgs (Dymond 83mins), Turgott, Goldberg (Porter 70mins). Subs not used: O'Connor, Huxter.
Welling: Lewington, Fish, Brown, Hatton, Hayles, Francis, Nanetti (Johnson 90mins), Waldren, Crawford, Coombes, Joseph-Dubois (Walker 78mins). Subs not used: Girling, Williams, Fuseini.
Attendance: 599.