Southwell scored a close-range header and a coolly-taken penalty to seal United's victory after Aaron Rey had earlier cancelled out Scott Garner's first half opener.
Obi Anoruo cut the deficit for Bay with virtually the last kick of the game, but the Pilgrims were soon able to celebrate a hat-trick of victories for the first time at the start of a season since 2010-2011.
PIC: Media Image.