The official announcement was delayed until Koeman returned from holiday.
“Ronald has such a strong track record in the game, instantly commanding respect for what he achieved as a player and for his qualities and accomplishments as a manager,” added Kenwright.
“We are really pleased to have secured the man who was our number one target from the moment we set out to appoint a new manager.”
Former Southampton goalkeeper Artur Boric, now at Premier League rivals Bournemouth, tweeted a Koeman quote from March which criticised a “lack of loyalty” in football.
Boruc added: “No words for this one.”
Southampton and England full-back Ryan Bertrand told BBC Sport: “The club is in safe hands, they’ve rebuilt on more than one occasion and I’m sure they’ll do it again.”
Saints finished seventh and sixth under Koeman – their best Premier League campaigns.
He is expected to be joined at Goodison Park by brother Erwin and fitness coach Jan Kluitenberg, although a deal for the pair is yet to be finalised.
Everton also considered Unai Emery, who has just left Spanish side Sevilla after winning the Europa League for the third successive season, and Manuel Pellegrini after he left Manchester City, but Koeman was at the top of Moshiri’s list.
Southampton said their search for a new manager has already started.
Koeman made more than 750 appearances as a player, scoring 253 goals, and won the 1988 European Championship with the Netherlands.
He also won four domestic league titles in both the Netherlands and Spain.
Koeman has also managed Vitesse Arnhem, Ajax, Benfica, PSV, Valencia, AZ Alkmaar and Feyenoord, winning the Dutch league with both Ajax and PSV, and a Spanish Cup with Valencia.
Phil McNulty, BBC Sport’s chief football writer:
Everton’s appointment of Ronald Koeman as manager is a significant statement of intent from new major shareholder Farhad Moshiri.
Koeman’s standing as a player for the Netherlands and Barcelona, for whom he scored the goal that won them their first European Cup in 1992, instantly elevates Everton’s credentials and credibility around Europe.
It is an indication of how Moshiri means business and Koeman will be backed by huge transfer funds this summer, irrespective of whether Romelu Lukaku and John Stones stay at Everton.
Moshiri wanted to make an appointment that would demonstrate the scale of his plans for Everton and Koeman fitted the bill – the idea of returning to David Moyes was never seriously contemplated despite the Scot being linked with a comeback at the club where he spent 11 years.
He was clearly happy at Southampton but the recent constant depatures of top players – as well as the personal and footballing finance on offer at Everton – swayed his thoughts towards Goodison Park.
It also means there is now a “northern powerhouse” of managers within 40 miles of each other with Koeman at Everton, Jurgen Klopp at Liverpool, Pep Guardiola at Manchester City and Jose Mourinho at Manchester United.
For Everton, however, the arrival of a character of Koeman’s stature will provide an immediate lift after the mutinous final days of the Martinez era.
Adam Blackmore, BBC Radio Solent Sports Editor:
Ronald Koeman did a fantastic job at Saints, especially given he came in at a time when the club were in meltdown following the departure of Mauricio Pochettino and a number of key players.
He is Southampton’s most successful manager in the Premier League era, and he managed the club with the same impressive calm authority he played with.
But his U-turn at St Mary’s has left a bad taste, and his surprise decision to swap the Europa League for a re-build at Goodison Park means his legacy at Saints will forever be badly tarnished.