Koeman took charge of Southampton last month and will be joined in the dugout by Lee, who serves as first-team assistant coach, while Watson joins as senior goalkeeping coach.
Lee was set to take a coaching role at Championship side Brighton, but changed his mind after he was approached by the Premier League club.
The 55-year-old spent 14 years in Liverpool’s backroom and most recently worked as head of academy coaching at Bolton Wanderers.
Watson is the current goalkeeping coach for the England senior team and moves to Saints after a two-year spell at Norwich.
The duo will link up with fellow coaches Erwin Koeman and Jan Kluitenberg, who joined Koeman at St Mary’s last month.
Executive director Les Reed told the club's website: “I am extremely happy that we have been able to appoint three highly-respected and talented coaches to these important positions in our coaching structure.
“Sammy has a wealth of experience from his time with Liverpool and Bolton, as well as from his long association with the England team.
“He will work closely with Ronald, Erwin and Jan, and will play a big part in the day-to-day coaching of the first team.
“Despite his relatively young age, Dave has already built up an impressive CV and is regarded as one of the most talented goalkeeping coaches in Europe, and has only just returned from the 2014 FIFA World Cup finals.
“That we have been able to attract Sammy and Dave to Southampton is testament to our ambition to continue growing as a club, and shows our commitment to develop talented footballers at all age levels.
“In addition, Martin is one of the most highly-qualified coaches in the English game and, after a successful time in charge of our under-21s, he will now play a big part in how coaching is delivered right from the first team down through the academy.
“He will work closely with the first-team coaching staff, but will play a wider role in ensuring that our teams play in the Southampton way at all levels.”
Southampton have also announced that Martin Hunter has been appointed as technical director after four years in charge of the Development Squad.