Southampton were expelled from the second-tier Championship play-offs last month after admitting to repeatedly spying on opponents' training sessions, and were given a four-point deduction for next season.
In its written reasons published on Monday, an arbitration panel laid bare the influence of Eckert in the scandal, saying it was "a contrived and determined plan from top down to gain a competitive advantage" and that analysts who carried out the unauthorised filming "felt pressurised to do the observations that Mr Eckert and the senior coaches wished them to do".
However, Solak said in a video message to fans published on Tuesday that the club believed Eckert "is the man to take us forward".
"As a board, we are fully behind him," the Serbian owner said, "and together we only have one objective - we want promotion back to (the) Premier League."
In an interview with the BBC, Solak said Eckert "deserves a second chance and I would give it to him".
Southampton qualified for the play-off final - after beating Middlesbrough, one of the clubs it spied on - before being expelled.
That deprived the Saints of the chance of promotion to the Premier League and a guaranteed windfall of at least $US270 million ($A376 million) in future earnings.
In a separate video message published by Southampton after Solak's, the 33-year-old Eckert said: "I am a young coach, I have made a mistake, and I take full responsibility.
"I am devastated," the German coach added, "that after six months of building that relationship (with the Southampton fans) back up, the season has come to an end - come to an end that couldn't have left us in a worse place than we are in right now."
In what he described as a "bitter irony," Eckert said: "None of what has happened had any effect on the sporting performance."
"I hope," he said, addressing Southampton's fans, "that over time you can understand and forgive."
