The deal will see the defender loaned back to Witton until the end of February.
Anson, 22, has made 33-appearances so far and has scored one goal.
Witton Albion Chairman John Salmon said: "Adam was a contract player with a release clause.
"The Southport FC offer exceeded this figure and, whilst we do not want to sell our best players, we will never stand in any player’s way if they want to play at a higher level.
"We thank Adam for his service and wish him all the best in his future career."
He will be available for their fixture tomorrow against Stafford Rangers.
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