Biggar scored the Ospreys' first try and booted four penalties and four conversions to pass 300 points for the season and 1,000 for the region, with further tries from Kahn Fotuali'i, Hanno Dirksen, Andrew Bishop and Rhys Webb completing a crushing semi-final success.
Munster had scored the first try through fly-half Ian Keatley, who also added the conversion and a penalty, but the Irish province were soundly beaten as boss Tony McGahan departed for his new role with Australia with a heavy defeat.
The Ospreys defensive effort was immense, with captain Alun Wyn Jones, Richard Hibbard and Ryan Jones leading from the front as they moved one win away from a fourth Celtic crown.
But their only error on that front allowed Munster to claim the opening try inside three minutes.
Keith Earls breached the midfield defence far too easily with his straightening run, and the Ireland three-quarter found Keatley to send the fly-half racing over for a score he also converted.
Biggar got the home side on the board with a ninth-minute penalty and they thought they had claimed a try of their own two minutes later, only for Shane Williams to be called back for a foot in touch after going over from Richard Fussell's pass.
But Munster hooker Mike Sherry's overthrow at the resulting line-out gave the hosts the position for Ryan Jones' neat offload out of the tackle to find Biggar's inside run for a try which the outside-half converted.
Keatley and Biggar traded penalties before Munster were forced into a reshuffle as Johne Murphy's ankle injury saw him replaced by Ronan O'Gara, with Keatley moving to the full-back slot.
This was proving to be a contest of ferocious intensity and some desperate, and illegal, ruck work under duress from Munster flanker Tommy O'Donnell allowed Biggar to put the Ospreys six points ahead with another penalty on the half-hour.
But having taken a stranglehold on the game, the Ospreys almost blundered their way into handing Munster back the lead.
Simon Zebo's chip ahead appeared harmless until Kahn Fotuali'i failed to gather, and when Ryan Jones took far too long to boot the ball clear, a chargedown handed Munster possession 10 yards from the hosts' line.
But they bravely held out the red tide and counter-punched with a thrilling long-range try from the last play of the half.
From turnover ball, Fussell spotted second row Mick O'Driscoll ahead of him and raced clear, profiting from a slip from Zebo, to link with Joe Bearman for Fotuali'i to cross beneath the posts for a converted score.
Munster's frustrations only increased after the interval as firstly Zebo spilt the ball during a promising attack, before their increasing inability to handle the Ospreys front row at scrum time allowed Biggar to slot his fourth penalty.
The Ospreys made the game safe with 54 minutes on the clock with another sensational counter-attack score.
Justin Tipuric's crunching tackle on O'Gara gained his side possession and when the ball was moved right, Beck and Fussell made telling thrusts for Dirksen to skate up the right touchline and give the hosts a 31-10 advantage.
Centre Bishop wriggled through a tackle under the posts for a fourth try after more strong work by Dirksen before replacement Webb rounded off a superb performance as he cantered over after Williams and Fussell had shredded Munster down the right flank.