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The new coach seemed set to triumph in his fourth match at the helm of the two-time European Cup winners thanks to a first-half brace from Igor Jesus, after Cedric Bakambu sent the Spanish side ahead.
However Antony popped up in the box to slot home in the 85th minute and claim Betis a share of the spoils.
Postecoglou, who won the Europa League last season with Tottenham before being sacked little over a fortnight later, is aiming to end Forest's wait since 1990 for a major trophy.
He dropped target man Chris Wood to the bench in favour of Jesus, and the striker signed this summer from Botafogo delivered for the Australian.
Betis' key man was former Manchester United winger Antony, he helped create the opening goal in the 15th minute through Bakambu and was the striker of ultimate level.
