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 City are grappling with ongoing injury worries ahead of Sunday's Premier League encounter with in-form Bournemouth at the Etihad Stadium, as Pep Guardiola navigates a mounting list of absentees.
A flurry of knockout drama unfolded in the Carabao Cup on Wednesday night as Arsenal edged Brighton, Palace stunned Liverpool, Manchester City rallied past Swansea, Chelsea survived Wolves, and Newcastle brushed aside Tottenham.
Arne Slot has confirmed that Liverpool defender Giovanni Leoni will be sidelined for around a year after tearing his anterior cruciate ligament in the midweek Carabao Cup tie against Southampton.
Liverpool's Carabao Cup win over Southampton came at a heavy cost as teenage defender Giovanni Leoni suffered a serious knee injury on his debut at Anfield.
