Woods, 39, makes the step up after two years working with the club's reserve and youth teams.
The first team have been without a specialist goalkeeping coach since Jim Blyth left Parkhead along with manager Gordon Strachan a month ago.
Woods told the club's official website: "I am absolutely delighted to take up this new position and I look forward to working with the first-team keepers at the club.
"I have been working at Celtic for some time and already know a number of the goalkeepers at Celtic.
"Having been part of the club, I am aware of the great demands and standards and I will be doing all I can to achieve these high standards and make as much of an impact as possible.
"We are fortunate to have top-class goalkeepers and it will be my objective to ensure that the guys are at the top of their game every week.
"This is a huge honour for me and I hope to play my part in bringing as much success as possible to Celtic."
Woods would appear to complete Mowbray's coaching team after he brought assistants Peter Grant and Mark Venus with him from West Brom.
He said: "We are delighted to make Stevie our new goalkeeping coach.
"I know that many people within the club have been highly impressed with Stevie's work in recent seasons and we are delighted to give him this opportunity to work with the first team.
"I have spoken to Stevie at some length about his new role. I too believe he has a lot to offer and I am sure he will make a great contribution to the first team."