Williams was denied a first-half try despite crossing the whitewash - the touch judge ruling his foot was out of play as he dashed into the corner - but with Munster, sorely missing absent captain Paul O'Connell, the Ospreys' fleetness of foot was too much for the Irish province to handle.
And it proved sweet revenge for the Ospreys, who were beaten by Munster 12 months ago at Thomond Park at the same stage of the competition, and who are now faced with Leinster or Glasgow in the final.
It was Munster who drew first blood as early as the third minute, Keith Earls making a fine break before off-loading to fly-half Ian Keatley who converted his own try for a 7-0 lead.
Biggar got the Ospreys up and running five minutes later with a typically clinical penalty before Williams' scorched over, only to see his effort ruled out.
But on 12 minutes, the hosts were in front when concerted pressure near the Munster try-line saw Ryan Jones barge his way through Earls before handing over to Biggar, who cruised over between the posts.
Keatley and Biggar then traded penalties to leave the Ospreys 13-10 up but with half-time approaching, the home side scored a try that will linger long in the memories of the Liberty Stadium faithful.
Having forced a turnover near their own line, the Ospreys resisted the temptation to kick for touch and burst down the right through full-back Richard Fussell, who fed the supporting Joe Bearman who in turn found Fotuali'i to score a contender for the RaboDirect PRO12 try of the season.
Biggar converted to give the Ospreys a 23-10 lead at the break and soon after the restart he scored the penalty that took him into four figures for the Welsh region.
Munster's set-piece, ordinarily so strong, was out of sorts without O'Connell marshalling but with or without their captain, they could do nothing about the pace of Ospreys' flying backs.
On 55 minutes the inevitable happened as Dirksen pinned back his ears and roared down the right to score in the corner while the Ospreys made the game safe when Bishop crashed his way over.
Munster were willing the final whistle to blow but replacement scrum-half Webb strolled over with ten minutes remaining to rub salt into their wounds as Tony McGahan endured a disappointing farewell.
|Ospreys Score Card|
|Munster Rugby Score Card|