Lyon claimed fourth-innings figures of 6-128 in Perth, ripping the heart out of the tourists on the final morning before they were dismissed for 333 in pursuit of 498 for victory.
It came as the men from the Caribbean lost four wickets in the first session on Sunday before Australia were forced toil harder for the final three in the session after lunch.
Australia did it without Pat Cummins in their bowling attack but he remained on the field with his quad injury despite being limited in movement.
How the world's top-ranked bowler recovers for the second final Test against West Indies in Adelaide on Thursday will now be a point of interest for the next three days.
Travis Head also claimed two wickets, including ending a 82-run seventh-wicket stand that had frustrated Australia when he bowled Alzarri Joseph for an entertaining 43.
Lyon then removed Roston Chase (55) in the deep and bowled No.11 Kemar Roach next ball, capping his heroics.
His haul took him past Ravichandran Ashwin (442) and into eighth position among Test wicket-takers overall, with 446.
Only Muttiah Muralitharan sits above the pair as the most successful offspinner in the game, with an untouchable 800 wickets to his name.
Lyon and 36-year-old Ashwin are set for a thrilling battle in India next February, with India to also play two Tests against Bangladesh before then.
Australia's finger spinner removed Kyle Mayers in the sixth over of the day when he had Kyle Mayers caught at slip, before spearing one in past Kraigg Brathwaite.
After the tourists' captain had reached his century on day four, Lyon went flatter and spun the ball back into Brathwaite's off stump on 110.
The spinner had earlier landed the other crucial blow late on day four, with Jermaine Blackwood caught at bat-pad on 24 when he and Brathwaite had built a 58-run stand.
Once derided for an apparent inability to close out games on day five late in his career, Lyon's haul also made him only the third Australian to pass 100 fourth-innings wickets, behind Shane Warne and Glenn McGrath.
The win is enough for Australia to retain the famous Frank Worrell Trophy, given they are the current holders and the series comprises only two Tests.
It also keeps alive the team's hopes of wrapping up a spot in the World Test Championship final before heading to India, with a faultless home summer possibly enough to do so.
Beyond Lyon's wickets on Sunday, the match will long belong to Marnus Labuschagne.
After his 204 and Steve Smith's 200no helped Australia's to 4(dec)-598 in the first innings, Labuschagne became only the eighth player in history to back up a double-century with a ton in the same match.
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