Ricky Gabriel was sent off in the 27th minute for Braintree but it took the Wings until the hour mark to make their man advantage count.
Goals from Callum Driver, Anthony Acheampong and Jack Jebb secured another three points for Welling - their third win in a row.
They’re now within two points of the play-offs and boss Jamie Coyle wants his side to use their momentum to push into the top seven.
Coyle told Wings TV: “It’s tough when you’re playing against ten men for so long. It’s difficult to break teams down but the professionalism and quality the boys showed was phenomenal.
“Braintree are a good side and one of the best footballing sides we’ve played this season. Our boys have really stuck together, got three wins on the bounce and it’s something we want to continue.
“It’s all about confidence and momentum. We looked really sharp and we can take loads of positives going forward.
“It’s the business end of the season now so we’ve scheduled another training session on a Monday. We’ll prepare well for Gloucester in two weeks’ time.”
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