Steve Cooper's Swans fell behind at the Liberty Stadium through Kamil Grosicki's third minute opener, with the Polish international's cross-shot flying past Woodman.
But after the restart, the Welsh side began in strong fashion with two headed goals in the space of 107 seconds, courtesy of Borja Baston and Mike van der Hoorn which secured three points in front of 15,741 spectators.
Elsewhere, midfielder Dan Barlaser played 69 minutes for Rotherham United as the Millers got off to a perfect start to their League One campaign, recording a 2-1 away win at AFC Wimbledon.
Freddie Ladapo, Rotherham's club-record signing, sent into the lead before the home side replied through Joe Pigott's header after the break but, with the League One encounter approaching its end at Kingsmeadow, Clark Robertson's late strike secured maximum points for Paul Warne's side.
Nathan Harker also made his debut for Lee Clark's Blyth Spartans on Saturday but was unable to record a winning start for the National League North club after suffering a 2-1 defeat to Gloucester City at Croft Park.