Higdon struck after 41 and 65 minutes as Saints claimed their first points on the road to deny United a third straight success.
Prince Buaben powered in an 86th-minute free-kick to set up a tense finale, but United suffered a third home loss of the campaign.
United were seeking to build on victories over Hibernian and Hamilton, but in a limp display failed to test Saints goalkeeper Craig Samson until Buaben's late strike.
Source: PA
Source: PA