Davis was on the brink of making a painful exit from the Betfred.com World Championship, trailing 12-4 to the man who has made a cruel habit of humiliating him on his favourite stage.
Robertson looked set to go through with a session to spare but still needs one more frame after Davis took the last two of the evening to postpone what is surely an inevitable defeat.
The 52-year-old Davis brought this tournament to life and won himself a new generation of supporters with his remarkable second-round victory against defending champion John Higgins.
It was tempting, and romantic, to believe amid the euphoria and the scale of his achievement that Davis might have rediscovered the form to take him all the way to the final, 21 years after the last of his six world titles.
But Davis warned he would be taking each ball and each frame as they came, rather than looking further ahead.
And in Robertson he met an Australian who potted ball after ball, winning frame after frame.
The 28-year-old from Melbourne showed Davis no mercy in the first round last year and repeated the treatment.
Davis gave another sell-out Crucible crowd a classic century to savour in the evening session, an immaculate 128, but that was out of keeping with the flow of the match.
Robertson raced into a 7-1 lead in the morning session and converted that into his emphatic overnight lead. The evening interval score of 10-2 had a familiar look, given that Robertson crushed Davis by the same margin 12 months ago.
The opportunity was there for Robertson to earn himself an unexpected day off, and that seemed likely when he edged their first frame back and then made it 12-2 with a sparkling 92 break.
Davis won the next frame on the black with a gutsy 32 clearance, and then crept over the line in the next, eventually potting the pink after a safety battle.
Robertson had packed his bags and cancelled his accommodation when he trailed 11-5 to Martin Gould in the second round. But after climbing out of that deep hole and dominating against Davis he has every chance of becoming the first man from outside the British Isles to land the world title since Canada’s Cliff Thorburn in 1980.
Robertson won the final session of that tussle with Gould 8-1 to edge through 13-12, and carried that form into his clash with Davis.