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.
VIEWS FROM THE DRESSING ROOM
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."
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