Barnsley loanee Mike Phenix's header put Southport ahead, while home goalkeeper Max Crocombe saved Scott Kashket's penalty.
Josh Thompson and Mark O'Brien made it 3-0 before Welling's Michael Bakare scored from 12 yards.
Welling completed the comeback with headers from Sahr Kabba and Ian Gayle and are eight points from safety.
Southport sit 17th, eight points clear of the relegation zone.
Welling came from 3-0 down and scored twice in in the last 10 minutes to secure a point at Southport.