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Jose Mourinho's side were 2-0 up from the first leg last week but collapsed monumentally in the second away from home this evening.
And Croatian international Orsic was the match-winner, producing a glorious clutch performance. He hit a hat-trick, including two beautiful goals from outside the box, to deliver a sensational comeback for Dinamo.
The 28-year-old struck twice in the second half to force extra time and then went and bagged the sumptuous solo winner in the 106th minute of action.
It completes a remarkable few days for the Croatian outfit, who saw their manager Zoran Mamic resign after he was sentenced to eight years and four months in prison for fraud and embezzlement.
"Although I don't feel guilty, I resign as I said before I would do if the sentence was confirmed. I wish all the good luck to the club," Mamic said in a statement.
Spurs piled on the pressure to find that all-important winner but managerless Zagreb hung on for a quite incredible victory and celebrated wildly at full-time.
Damir Krznar, a former Dinamo player, took charge of the fixture and masterminded one of the club's greatest ever European results.
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