Within only four hours, Australia went from being on track to a probable defeat to pulling off a famous win by the end of the day.
Australia needed to make the biggest total of the match to win, but Head did it with ease as he blasted England's express quicks to all parts of Perth Stadium following his unexpected promotion to open.
Player of the series when England last toured Down Under in 2021-22, Head smashed his 10th Test century from only 69 balls - the fastest fourth-innings ton in history and equal third-quickest by an Australian, only behind Adam Gilchrist and Jack Gregory.
With just 13 runs needed, Head was finally out for 123 from only 83 balls when he was caught on the boundary off Brydon Carse's bowling.
Only one other player, Harry Brook (52 in England's first innings), scored a fifty in the match.
Head's move from No.5 to partner debutant opener Jake Weatherald only came about because veteran Usman Khawaja continued suffering the back spasms that kept him off the field on Friday.
The 31-year-old opens in white-ball cricket, and has done the job in Tests in Asia, but Australia have been reluctant to use him there at home.
Cruising at 1-65 with a lead of 105 just after lunch on day two, England looked headed to snap a 15-Test winless streak in Australia dating back to January 2011.
But England suffered a crushing collapse of 6-39 during an 11-over period, losing 9-105 during the second session to be out for 164 in 34.4 overs.
Their two innings in the match lasted just 67.5 overs, making it the quickest they have been bowled out twice in any Test since 1904.
"This is going to hurt England, it's going to damage them I think," England's Ashes-winning captain Michael Vaughan told Kayo.
"They have the tools to be really competitive, but you can't be competitive without brains.
"You can't just play one way, which is what they've done, and they've come unstuck."
Chasing 205 to win, Australia exploded out of the blocks, particularly given the first three innings of the match had started 1-0 for the first time in Test history.
Head and Weatherald, who was out for a second-ball duck in the first innings, took England's quicks on with a rapid-fire 75-run opening partnership.
Weatherald, who was cheered when he finally got off the mark, was out for 23, but the recalled Marnus Labuschagne (51 not out) was the ultimate wingman for Head.
Stand-in captain Steve Smith arrived at the crease to help Labuschagne tick off the remaining runs.
Star Australia quick Mitchell Starc achieved a 10-wicket match haul for the third time in his career, adding three second-innings scalps after his career-best 7-58 on Friday.
In frightening news for battered England, Australia captain Pat Cummins is a strong chance to return for the second Test at the Gabba.
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