Saints were thrown out of the Championship play-offs last month after admitting sending a young intern to spy on a Middlesbrough training session.
The English Football League said the south-coast club were found guilty of a "deplorable" act of putting pressure on the junior member of staff to spy on Boro.
In a damning verdict, the governing body said Southampton boss Eckert authorised the spying.
The German, 33, has now put out a lengthy video statement on club media, which began: "I will try to be as honest and clear as I can be.
"For everything that has happened, I do want to apologise and I hold my hand up because as head coach I am responsible.
"I am responsible for everything that has happened at this football club."
Southampton owner Dragan Solak said he was standing by Eckert.
The Serbian media mogul admitted Eckert had made a "mistake", but was adamant it should not cost a "super-talented manager" his job.
Solak told the BBC: "I have to believe, honestly, and I believe Tonda, that he didn't know it was the rule that he was breaking."
He added: "My personal opinion, and the opinion of the board, is that he is a manager who deserves to be backed by us and to be supported by us.
"I think he deserves a second chance and I would give it to him. My first support would be behind him, actually, because I think he is a super-talented manager."
Southampton will also be docked four points next season after admitting to multiple breaches of regulations related to the "unauthorised filming of other clubs' training" sessions.
Eckert is not yet totally out of the woods as the Football Association (FA) has launched a probe into the affair and could still charge him, meaning he may face a ban if found guilty.
Solak, though, believes that Eckert has been subjected to "a witch-hunt" and the club has been "over-sentenced".
"My support comes from a very simple legal situation where there is no double jeopardy," said Solak.
"Whatever crime you did, you can be sentenced only once."
Middlesbrough, who lost to Southampton in the play-off semi-finals, were reinstated for the final at Wembley on May 23rd but were defeated by Hull, who will join Coventry and Ipswich in the Premier League next year.