"At certain moments in your life you have to make a decision and I am almost 68 years old now. A number of clubs contacted me but the decision was always Sunderland or nobody," Advocaat said in a statement on the Sunderland website (www.safc.com).
"We gave everything in our time there and achieved what we set out to. Sunderland has been one of the highlights of my career."
Advocaat took over at the Stadium of Light in March until the end of the season, replacing the sacked Gus Poyet.
The Black Cats were one point above the relegation zone when he arrived amid some disarray but the experienced manager led them to 16th, three points above the condemned clubs.
Short added: "Although I am extremely disappointed, I understand and absolutely respect his decision.
"In his brief time here he made a real impact and lifted the club with his experience and enthusiasm. We will always be grateful to him for what he achieved."
The defining image was Advocaat weeping with joy when the club secured survival with a goalless draw at Arsenal in their penultimate game of the season.
(Writing by Ian Chadband; Editing by Ken Ferris)
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