Two minutes later the Ospreys were level as Biggar dropped a goal on their first attacking visit to Leicester's 22.
The tit-for-tat shots at goal continued as Flood sent over his second penalty in the 12th minute, Biggar levelling a minute later.
Centre Jeremy Staunton matched Biggar's drop-goal effort in the 23rd minute to put Leicester back ahead and the home fly-half again levelled in the 25th.
But all those kicks at goal merely masked the tussle up front in which Leicester's forwards found their match among the home pack.
The scrums were hotly contested and often ended collapsed, much to the frustration of Irish referee Alan Lewis.
But both sides exercised control of their own line-outs, leaving the rival backs to play their own cat-and-mouse games.
At half-time the Ospreys could claim to have come out on top in that area, courtesy of the only try thus far.
At fly-half Biggar was a model of calm assurance, but the man who would also like to wear the Ospreys' number 10 jersey, James Hook, provided the moment of magic that brought the try.
Hook's clever and expertly-placed and timed grubber kick gave wing Bowe the chance to beat the defence to gather and score for a try confirmed by the video ref.
Biggar failed with the conversion, leaving the Ospreys 14-9 ahead at the break.
By that time Clermont Auvergne had effectively won Pool Three by scoring a bonus-point four tries in what ended as a 59-20 win at Viadana.
Wales fly-half Biggar also failed with a penalty and drop-goal attempt in the 43rd and 44th minutes, while Byrne failed with a long-range drop-goal in between Biggar's attempts.
Biggar and Byrne could then only watch as Flood kicked a 50th-minute penalty and after referee Lewis expressed more unhappiness with the scrummaging of rivals Dan Cole and Paul James, Leicester replaced the former with Martin Castrogiovanni.
Biggar put his side five points ahead in the 55th minute and the fresh blood in the Tigers' front-row could not prevent the Ospreys demolishing the visitors' scrum in the 63rd.
There was controversy at the start of the last quarter as Byrne departed for treatment to a cut and was replaced by Sonny Parker.
When the Wales full-back returned no-one was removed from the hosts' ranks, which meant they had 16 men on the field for just less than a minute.
Byrne was then removed from the field once more and when Biggar finally made way in the 73rd minute, Byrne returned to the fray legally.
Leicester launched a series of increasingly desperate attacks as the clock wound down, but replacement hooker George Chuter knocked on in the 79th minute and the visitors conceded a penalty at the scrum in the shadow of the hosts' posts.
The Ospreys cleared their lines and replacement Shervington found fellow replacement Filo Tiatia at the line-out before Januarie's kick to touch ended the Tigers' hopes.
It was also the final moment of Januarie's three-month Ospreys stint before he returns to Super 14 side the Stormers in his homeland.
Ospreys boss Scott Johnson: "We are glad to get out of this group.
"There is a great history in sport of sides being successful who come out of tough pools.
"The defence in the last 15 minutes was as tough as anything you would get in international rugby.
"The last two weeks (Ospreys have played Leicester and Clermont) have been of Test match intensity."
Leicester boss Richard Cockerill: "We are out of the tournament. We have to go back, dust ourselves off and cope with the adversity of losing.
"It was a good contest, but someone has got to lose, and it was us today. Sometimes you get the luck, sometimes you don't.
"Ospreys are a good side with quality throughout. They've got the personnel to win the tournament, but there are some good teams left and Clermont would probably be my favourites."
Ospreys: Byrne; Bowe, Bishop, Hook, Williams; Biggar, Januarie; James, Hibbard, Adam Jones, Alun-Wyn Jones, J Thomas, Collins, Holah, R Jones. Replacements: Walker for Williams (79), Parker for Biggar (73), Bennett for Hibbard (76), Gough for J. Thomas (66), Tiatia for Collins (76). Not Used: Bevington, Griffiths, Nutbrown.
Leicester: Hamilton; J Murphy, Hipkiss, Staunton, Tuqiri; Flood, B Youngs, Ayerza, Davies, Cole, L Deacon, Parling, Newby, Moody, Crane. Replacements: Tuilagi for J. Murphy (56), Mauger for Staunton (62), Stankovich for Ayerza (79), Chuter for Davies (67), Castrogiovanni for Cole (53), Woods for Newby (60). Not Used: Kay, Grindal.
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