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Welling remain deep in relegation trouble after failing to score against the division's tightest defence on Tuesday night.
The Wings even had the advantage of playing against 10 men after Braintree's Sean Marks saw red for a blatant elbow on Welling centre-back Ian Gayle.
Braintree took a ninth-minute lead after a dreadful mistake by Michael Chambers. Remy Clerima's cross should have been cleared easily, instead Chambers sliced the ball to the far post for Dan Sparkes to tap home.
Welling nearly responded immediately as Braintree keeper Michael Crowe saw his clearance fall to Xavier Vidal, whose shot was cleared off the line by Mark Phillips.
Kadell Daniel looked Welling's brightest hope going forward and he had two long-range shots saved by Crowe.
Braintree were correctly reduced to 10 men on 33 minutes when Marks elbowed on-loan Gayle, who was making his debut. After communication with his assistant, referee Dean Treleaven brandished a red card.
Gayle could no longer continued and was replaced by Nortei Nortey, who hardly put a foot wrong in the centre-back position for the second successive game.
Welling almost equalised in stoppage time when skipper Harry Lee let fly from at least 35 yards but his left-foot rocket came back off the angle of post and bar.
The visitors struggled for any momentum in the second half and it was Braintree who created the better chances.
Welling introduced Luke Wanadio for his debut, moments after Michael Bakare had seen his left-foot shot come back off the post - the second time in the match Welling hit the woodwork.
But that was as close as they got as Braintree survived quite comfortably in the end with Welling seemingly reluctant to put more bodies forward.