A breathtaking final quarter saw the Ospreys come from behind to secure a crucial bonus point win, the fourth try coming with time up, to cement their place in the play-off spots.
The region overcame two questionable second half yellow cards to record a memorable success at Stadio Comunale di Monigo that moves them up to second place in the PRO12, although both Ulster and Munster could sneak back ahead on Saturday.
There were tries from Eli Walker, Dan Evans and Justin Tipuric, before Rhys Webb went over in the final play of the game to secure the bonus point with his ninth PRO12 try of the season – a new Ospreys record.
Some bright attacking play from the Ospreys, Tipuric, Webb and Walker all carrying well and showing neat hands, was rewarded with an early penalty in front of the posts after Treviso were guilty of killing it on the ground. It was a simple kick for Biggar and he gave his team the lead with just a little over two minutes gone.
The Ospreys were on top in the early exchanges as they looked to play a fast, running game, but on more than one occasion in the first quarter they paid the price for a lack of accuracy and patience, losing possession when close to the Treviso line.
Having spent the best part of 20 minutes in their own half, Treviso made a rare foray into Ospreys territory at the midway point of the first half, Jayden Hayward sparking the move with some strong running, and when Tipuric was penalised for hands in the ruck, it was Hayward who leveled the score at three points apiece.
The game then settled back into the same pattern, the Ospreys pressing and Treviso defending resolutely, but after referee Dudley Phillips issued a general warning to both teams about illegal hands in the ruck, Enrico Bacchin was shown yellow on 25 minutes for doing just that. The penalty allowed Biggar to reclaim the lead for the visitors.
It was proving to be a frustrating evening for the Ospreys as they struggled to break down their opponents, even with the extra man, Bacchin returning to the field with no change to the score.
They finally managed to carve open the defence in the final seconds of the half, patiently going through the phases from a scrum slap bang in the middle of the Treviso half.
Dan Baker, Josh Matavesi and Tyler Ardron all made the hard yards to suck in defenders before Walker marked his 50th appearance for the region by cutting in from the right touchline, sidestepping two defenders before shrugging off Hayward to score.
Biggar’s conversion came back off the upright, meaning the Ospreys went in at the break with an eight-point lead.
HALF-TIME: BENETTON TREVISO 3 OSPREYS 11
The Ospreys found themselves reduced to 14 men for 10 minutes within 90 seconds of the restart. It was Nicky Smith, incredibly, who found himself sent to the bin after being penalised at his first scrum after replacing Ryan Bevington at the start of the second half, clearly paying the price for the troubles his colleagues had endured at the set-piece in the first half.
With the numerical advantage, Treviso were able to enjoy their first spell of prolonged pressure, enjoying more than 30 phases of possession inside Ospreys territory.
Their patience was eventually rewarded when it was spun wide to the wing where Simone Ragusi, who spent a year with the Ospreys as a teenager in 2011/12, was able to touchdown in the corner despite the attentions of Evans, Tom Grabham and Jonathan Spratt.
Hayward’s conversion meant that Treviso were back within a point by the time Smith returned to the field but, almost immediately, the Ospreys were once more down to 14, Webb sinbinned for what Mr Phillips somehow ruled to be a high tackle on Ragusi.
Treviso full back Hayward made the Ospreys pay the full price for the yellow, slotting over three points to put his team ahead for the first time.
The Ospreys were in danger of losing their cool as frustrations at some of the refereeing decisions were growing.
Biggar was just off target with a long-range effort just after the hour mark, but they saw out the sinbinning without any further setback meaning they were still just two behind when Webb returned to the field.
Almost immediately, the Ospreys were back in front after Evans went over for his sixth try of the campaign. Walker, the scorer of the first Ospreys try, was the creator with a great run up the left flank before Ludo Nitoglia hauled him down just metres short. Evans was up in support though and he reacted well to snipe over from close range. Biggar again missed the kick, which was from an identical position to the penalty minutes earlier, close to the left hand touchline.
Biggar did find the target in the 72nd minute, his kick from 10m further upfield and infield giving the Ospreys a little breathing space.
The game then descended into farce as the referee and TMO tried to identify, without success, who they felt was guilty of taking out Biggar in the air as he collected his own up and under.
The win was secured with just a little over two minutes remaining, some great work from Ben John, on for Spratt, and Webb creating the opening for Tipuric to go under the posts and score. Despite time almost being up, the Ospreys were sniffing a crucial bonus point, and Biggar quickly drop kicked the conversion to give it one more go.
They were duly rewarded as, with time up, Matavesi supplied Grabham on the right and the winger carried well before feeding Webb who scampered over for the crucial fourth try, his record breaking ninth of the season in the PRO12.
Biggar’s conversion rounded off what could turn out to be a crucial victory for the Ospreys, a win that guarantees they finish as top Welsh region to secure their spot in next season's European Rugby Champions Cup as they hunt down a place in the play-offs.