Welling's Joe Healy missed a late penalty as Salisbury took a point after playing for 45 minutes with 10 men.
The midfielder put his spot-kick wide in injury time after goalkeeper Will Puddy had brought down Doug Bergqvist.
Puddy earlier made three saves before Theo Lewis saw red for a bad tackle on Kurtis Guthrie just before the break.
Wings striker Loick Pires then had a header ruled out for offside, while the Whites' best chance fell to Ben Wright who struck the bar from 12 yards.
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Salisbury City manager Mikey Harris told BBC Wiltshire:
"It's a fantastic point. I'm so proud of them today.
"To go down to 10 men before half-time for a sending-off which was never a sending-off - the way we dug in and worked for each other was a fantastic achievement.
"We had our chances to win it as well. We've taken four points over the weekend and that's back-to-back clean sheets."