Strachan quit a day after a 0-0 draw with Hearts heralded a disappointing end to an otherwise successful four years in charge.
Strachan was never fully accepted by a large number of Celtic fans and he never hid his unease at the media coverage surrounding the Old Firm.
The Celtic board will now need to act fast to appoint a new manager, whose season will begin in late July with a Champions League qualifier.
West Brom manager Tony Mowbray, a former Celtic player, is among the early favourites. Owen Coyle’s stock has risen considerably while Motherwell boss Mark McGhee has long been touted as a candidate after leading the Steelmen back into Europe. Dundee United manager Craig Levein is also highly regarded.
Strachan’s successor will have a hard act to follow. The former Coventry and Southampton manager became the first Celtic manager to claim a hat-trick of titles since Jock Stein, despite having less money to spend than predecessor Martin O’Neill.
Strachan declared it an “honour” to have managed Celtic.
He said: “I have enjoyed my time immensely at Celtic and clearly I will be sad to leave this great club.
“As I have said before, Celtic is a special club, with special supporters.
“There is no club like it and it has been an absolute honour and privilege to be the club’s manager for the past four years. We have done all we can to bring success to the club and we were delighted to deliver three SPL titles, other domestic silverware, as well as making some great progress in Europe. It is just disappointing that we did not manage the fourth championship.”
“I wish Celtic and our supporters nothing but success for the future.”
Main shareholder Dermot Desmond recognised Strachan’s part in reducing club debt and wages.
“Gordon leaves Celtic a better club both on and off the balance sheet,” Desmond said. “I am extremely grateful for his contribution.”