Djokovic, bidding to become just the second man in history to hold all Grand Slam titles at the same time twice, was trailing 1-4 in the final set and was 40-40 when play was halted.
Thiem led 6-2, 3-6, 3-1 overnight when play was suspended due to rain and the threat of high winds, before claiming the third set 7-5.
Djokovic levelled the match by taking the fourth set 7-5.
Organisers said the semi-final would not resume until 1300 GMT (15:00) at the earliest, which was the original scheduled time for the women's final between Ashleigh Barty and Marketa Vondrousova.
The winner will face 11-time winner and defending champion Rafael Nadal in Sunday's final.
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