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Staunton will be assisted by former England manager Bobby Robson, who has signed a two-year deal as International Football Consultant, with the position to be reviewed at the end of the term.
Staunton's backroom team includes Kevin MacDonald as coach, with the 45-year-old retaining his role as Aston Villa reserve-team manager, and former Republic number one Alan Kelly as goalkeeping coach.
For the supporters, Staunton is not the big-name appointment they were hoping for as Martin O'Neill, Alex Ferguson and Terry Venables had all previously been linked with the post.
Instead, many fans have been expressing their dismay, yet Delaney is adamant after a three-month search they have the right man.
All he is demanding is loyalty and faith, as he told PA Sport: "Managing the Irish team is a hugely high-profile role. Everybody in Ireland has a view on who the manager should be, so whoever it was there was going to be different views. No question about that.
"But the feedback I've had has been terrific, with people saying 'on you go'. They are delighted with it. The sub-committee of three who sat in judgment were unanimous and we're also delighted, while the board of the Association are also thrilled."
He added: "What is important now, after making this appointment, is everybody gets behind Steve to bring the good times back to Irish football. I'm certain he is the man to do it, so let's get behind him. That's my message to the Irish public."
Delaney has insisted it was Staunton's decision to bring in Robson, with his years of management experience certain to prove invaluable.
"Steve came to us with his management team, Kevin, Alan and Bobby," added Delaney.
"Bobby brings huge qualities in assisting and supporting Steve in whatever way he needs to get the best out of this Irish team. He took England to a World Cup semi-final, managed in Europe with some of the best clubs. You can only learn from somebody like Sir Bobby Robson."
He continued: "When I saw his name as one of the backroom team I was delighted. Steve's recommendation didn't surprise me."
Staunton, whose only coaching experience has been in recent months with Coca-Cola League One Walsall, insists he has what it takes as a leader.
Staunton and Robson have just 45 days to prepare for their first match, with Sweden due at Lansdowne Road for a friendly on March 1.