With a six-lead over Rangers heading into the penultimate match of the season, Celtic only needed to draw to seal the title. After drawing blank in a dominant first half at Tannadice, the visiting champion-elect took the lead through Giorgos Giakoumakis midway through the second period.
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The advantage was short-lived, as a more spirited Dundee United first hit the post through 16 year-old striker Rory Macleod, then draw level via a stunning strike from Manchester United loanee Dylan Levitt.
His goal set up a nervy finish for Celtic, but the visitor held on to confirm their return to the pinnacle of Scottish football.
Postecoglou, who guided teams to domestic titles in Australia and Japan prior to his appointment as Celtic manager in June 2021, will get the chance to lift another piece of silverware when Celtic is presented with the trophy after its home clash against Motherwell on Saturday (AEST).