Phil McGuire gave Teddington the lead after 30 minutes, but with John Shaw showing touches of brilliance, Danny Kerry and Pietro Attala converted second-half goals for Southgate to secure a well-deserved victory.
Hounslow's Rob Thompson scored three times in the first televised match of the season at Birmingham University as the champions demolished newly promoted Reading 5-1.
Thompson, however, was outdone by Neil Barker who scored five in Old Loughtonians 7-0 rout of Welton.
Havant gave notice of their intentions with a 4-0 win at St Albans while, in the Second Division, newly promoted Oxford University won 6-3 at Harborne.
In the women's National League, the Premier Division champions Ipswich stay top, ahead of Slough and Leicester on goal difference. Ipswich's winning goal against Clifton came from Vickey Dixon.
Slough's 2-0 win against Chelmsford was achieved through goals by their captain, Karen Brown, and Kate White, while Leicester, with second-half goals from Kim Gordon and Morag Laird, came from behind to win 2-1 at Ealing.Reuse content