Munster lost Tommy O'Donnell to the bin just after half-time, during which time Ospreys scored a decisive penalty try, before Damien Varley was sent off for a kick five minutes from time.
The opening ten minutes were scoreless with both defences well set in the face of early pressure before Munster conceded two penalties at the breakdown in quick succession.
Biggar coolly converted both as the champions found their stride - adding the first try of the match ten minutes later after a good spell of pressure.
Andy Bishop started the move after finding a gap off a line-out, with the ball moving to Ashley Beck before Fussell evaded the tackle of Keith Earls to go over.
After Biggar's fantastic touchline conversion gave Ospreys a 13-0 lead, Ronan O'Gara responded immediately with a sweetly-struck penalty to cut the deficit to ten points.
Munster slowly started to come into the game but were struggling to match Ospreys at the breakdown, though Biggar was unable to take advantage after curling a penalty wide of the upright on the half-hour mark.
The southern province side were pushing for a try and come close to scoring in the corner five minutes from half-time but Denis Hurley knocked on as the red shirts stormed towards the try line.
The visitors had to settle for an O'Gara penalty in front of the posts, before he added a third after the break when Ryan Jones was penalised for an infringement at the breakdown.
Munster were in the ascendency but lost Tommy O'Donnell to the bin two minutes after O'Gara's penalty, the flanker collapsing a maul as Ospreys pushed towards the 22.
Biggar added the three points to make it 16-9 as the champions looked to make the most of their numerical advantage.
They did just that seven minutes into O'Donnell's time on the sidelines, awarded a penalty try after Munster were penalised for repeatedly collapsing the scrum three metres from their line.
O'Gara responded with the boot again on the hour mark before Rob Penney made a raft of changes to try and get Munster back into the game.
But it was Ospreys who nearly scored, Keith Earls scrambling back to haul Fussell down metres from the line after the full-back capitalised on a loose ball.
Munster's replacements looked to have given the visitors a new lease of life and they soon closed the gap to eight points, replacement Ian Keatley finding target.
And just as the visitors looked like they could find a way back into the game Ospreys grabbed the vital try eleven minutes from time.
A brilliant run from Justin Tipuric down the left wing opened up the Munster backline and, after going through another three phases prop Hibbard was the unlikely recipient out wide for an easy run across the line.
The night went from bad to worse for the visitors six minutes from time as hooker Damian Varley saw red for kicking.
Ospreys pushed for the bonus-point try in the closing stages but had to settle for the 30-15 victory, a perfect way to continue their momentum after last weekend's victory over the Scarlets.
|Ospreys Score Card|
|Munster Rugby Score Card|