The Ospreys had a gilt-edged early chance after Mike Phillips launched an attack by gathering Biggar's kick-off.
Richard Hibbard bounced through two tackles to help take the Welsh side close, then Marty Holah was held up.
But a metre out Phillips tried to force his way over as Hibbard stood free a pass away and the chance was gone.
The Ospreys may have viewed that opportunity as merely a hint of what was to come. Instead the rest of the opening period largely belonged to London Irish.
Fly-half Daniel Bowden and opposite number Dan Biggar exchanged penalties, but the host pivot enjoyed the better of the attacking duel.
Bowden gave that pass that allowed full-back Delon Armitage to drift past Holah and make the run that led to Tagicakibau going over wide out.
The Exiles fly-half was unable to convert, but made amends with a 28th-minute penalty, then he and Biggar again exchanged successful shots at goal to leave the hosts 14-6 down at the break.
Even worse followed at the break. Wales and British and Irish Lions prop Adam Jones had his left elbow strapped up following an injury at a 25th-minute scrum collapse.
And the tight-head was replaced at the break by Craig Mitchell.
The Ospreys' dejection was summed up in Mike Phillips' expression
After Elvis Seveali'i broke through Bowden added to the Ospreys woes by sending wing Topsy Ojo over two minutes into the second period.
Biggar soon struck back, but missed another kickable penalty to leave his side 12-points down with 25 minutes to go.
Biggar departed amid a rash of changes that also involved Justin Tipuric coming into the back-row and Hook moving to the fly-half slot.
Those changes allied to the Ospreys' increasingly desperate attacking attempts gave Hook the chance to cut the deficit with a penalty in the 66th minute.
Bowden had the chance to restore the Exiles' advantage, but his kick went wide. And after a sustained period of pressure from a close-range line-out the hosts saw hooker James Buckland sin-binned for the latest in a series of Irish offences.
What followed summed up the previous 73 minutes. The Ospreys created the space to score, but under pressure from the Exiles defence Sonny Parker was denied then Mapasua cut down Jerry Collins with a tackle that led to a penalty that allowed the hosts to clear their lines.
Ryan Lamb, who had replaced Bowden, drove the final nail into Ospreys hopes with a late penalty.
London Irish boss Toby Booth: What delighted me was the effort, but then I was proud of the effort in the Saracens game and the Quins game. This was a win, but then it was achieved in the Heineken Cup.
"We came under the cosh from the very kick-off, and the graft of the tight five was absolutely outstanding. I am so pleased for the players.
"It was crucial for us to get the result and we've got it. We are now one from one.
"You have to remember why you play the game. We quoted Spurs boss Harry Redknapp, who says nothing beats winning, and that is so true.
"The monkey is off our backs and we go to Munster to face a team with such a huge tradition, and with a desire to make amends for their defeat in France."
Ospreys head coach Sean Holley:
"Failing to get over the line in the first minute set the trend, and we did that too often.
"We came up against a pretty desperate Irish defence, a team desperate to win, a team needing a win badly, and we dropped too many balls, made too many mistakes.
"Against guys such as their backs, you dare not miss touch because they will punish you. We had possession, we had periods when we should have made scores.
"This has been a brutal pool, and it's actually sad when three very good sides go out of the competition because of the way we have all damaged each other.
"But we have a very diligent coaching team, we have young talent coming through, and I am convinced that the future is bright.
"We have a strong spine in our side, and it's a fact that this is a very hard competition to win. We'll bounce back as best we can, but the saddest aspect of this is that we probably won't get a Welsh team in the quarter-finals.
"This is a huge opportunity lost, but you have to cope with that in a professional sport."
London Irish: D Armitage; Ojo, Seveali'i, Mapusua (capt), Tagicakibau; Bowden, Allinson; Corbisiero, Buckland, Rautenbach, Kennedy, Garvey, Stowers, S Armitage, Hala'ufia. Replacements: Thompstone for Tagicakibau (79), Lamb for Bowden (74), Murphy for Corbisiero (70), Ion for Rautenbach (53), Gibson for Stowers (70), Blaney for S Armitage (72).
Not Used: Casey, Lalanne.
Sin Bin: Buckland (71).
Ospreys: B Davies; N Walker, Bowe, Hook, Fussell; Biggar, M Phillips; James, Hibbard, A Jones, R Jones, A W Jones (capt), Collins, Holah, Thomas. Replacements: Byrne for B Davies (53), Parker for Biggar (59), Nutbrown for M Phillips (76), D. Jones for James (59), Bennett for Hibbard (59), Mitchell for A Jones (40), Tipuric for Holah (50).
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