Saints seemed in control when Tyreke Johnson and Marcus Barnes put them ahead early in the first half.
But Kourtney Hause pulled one back just before the break and Wolves captain Wilson equalised immediately after the interval.
Wilson then put the visitors ahead on 57 minutes before completing his treble in stoppage time.
Wolves overtook Saints and Newcastle United to climb into seventh, while the result stretched Southampton's winless PL2 run to five matches.
See: Southampton report
Southampton: 18 Dec v Reading (A) PL2
Wolves: 18 Dec v Blackburn (H) PL2