The England Under-21 international has signed a four-year deal at St Mary's and is reunited with Russ Martin, who he played under at the Welsh club.
Wood, 22, said: "It's obviously a massive club, a Premier League club that was in the Championship last season, and to get straight back into the Prem is a big thing. I'm very happy to be here.
"It's always been an attractive club. Obviously the style of football is a big thing, the coaching is a big thing, and it's a step up in leagues for me as well.
"It's a no-brainer really when you put all those things together. It's quite an easy decision to make."
Martin added: "Nathan is a player I know and trust from our time together at Swansea, and someone I also tried to sign in my first job at MK Dons.
"He's a top defender with tremendous physical qualities, great pace and calmness on the ball."
Brighton have signed Netherlands international Mats Wieffer on a five-year deal.
The 24-year-old defensive midfielder joins from Dutch side Feyenoord.
Technical director David Weir said: "We're delighted to welcome Mats to the club, a player whose qualities we have admired for a long time.
"He has a lot of experience in the Champions League as well as the Eredivisie and has developed into one of the best deep-lying midfielders in Europe.
"We obviously have Dutch players here already in Bart Verbruggen, Jan Paul van Hecke and Joel Veltman and I'm sure they will help Mats settle into the club.
"We're looking forward to helping Mats' career develop with Brighton."
Manchester United have strengthened their goalkeeping department after completing the signing of Wales Under-21 international Kit Margetson from Swansea City, with the move subject to registration.
Pep Guardiola has urged restraint despite Manchester City moving within touching distance of another historic domestic sweep, insisting talk of a treble remains premature even after they reached a fourth consecutive FA Cup final.
Nico Gonzalez struck a late winner as Manchester City secured a thrilling 2-1 victory over Southampton to reach a fourth consecutive FA Cup final at Wembley Stadium on Saturday evening.
Pep Guardiola has confirmed Rodri will not feature in FA Cup semi-final action against Southampton at Wembley on Saturday, with Manchester City opting for caution as the midfielder continues his recovery from a groin injury.
