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Lee Clarke took his Blue Square South tally for the season to 16 as Welling ended a run of four straight defeats on Saturday.
Clarke, the division's player-of-the-month for March, has been Welling's stand-out performer as they dragged themselves away from the relegation zone and mounted an ultimately unsuccessful play-off charge under new boss Jamie Day.
With Day's squad playing for new contracts, Clarke did his cause no harm after a poor first-half when neither side got going.
Weston took the lead early in the second period through Ben Cleverley.
But Welling responded when Clarke slotted home a penalty and he netted the winner with 15 minutes left.
Welling now know that victory at home to Basingstoke in their final game next Saturday will guarantee them a top 10 finish.
Bromley's last home game of the season finished with a whimper after a 2-1 defeat to Bath City.
Kaid Mohamed put the play-off chasing visitors ahead with an overhead kick on 17 minutes. The striker had been denied by the crossbar just a few minutes earlier. Bromley pulled themselves back level after 30 minutes through an own goal after Steve Ferguson's corner took a deflection.
The men from the West Country regained their lead on 68 minutes as Lewis Hogg's free kick was eventually guided home by Chris Holland.
Cray Wanderers still have a nervous end to the season in the Ryman Premier League after a 2-0 defeat at Carshalton.
Pico Gomez and Barry Stevens scored in either half for the hosts, with George Porter going closest for Cray with an effort against the post at 1-0.
Cray could still be sucked into the relegation zone after Maidstone beat fellow strugglers Ashford Town (Middx).