The fly-half stepped up with the clock having hit 80 and landed a tricky penalty to put the seal on a 35-32 success.
It was an incredible moment at the end of an incredible match at Stadio Comunale di Monigo, where Saints had fallen 25-8 down.
They roared back during the second period, scoring 24 unanswered points before Benetton drew level with nine minutes to go.
It looked like the game would end in a draw when Jamie Gibson delivered a crucial turnover inside his own 22, but Saints still had time to mount one last attack.
And after wing Ratuva Tavuyara was penalised for a deliberate knock-on, Biggar bisected the posts brilliantly to give his side a superb away win.
It means Saints have now collected nine points from their opening two Champions Cup games ahead of a big December double-header against Leinster.
Chris Boyd's side had headed to Italy on the back of a 25-14 win at home to Lyon and Biggar, who notched 20 points in that win, was on target early on against Benetton.
But the hosts were quick to bite back, showing the same ambition they had displayed at Leinster on the previous Saturday, putting a kickable penalty into the corner.
And eventually hooker Hame Faiva forced his way over the line, with Tommaso Allan adding the conversion to make it 7-3 to the hosts.
But Saints issued the response they needed, immediately putting pressure on Benetton and producing a sublime move that ended with Tom Collins scoring in the corner.
Biggar, whose brilliant pass opened the door for the try, was unable to convert from the touchline.
And Benetton were soon back in front as Allan landed his first penalty of the game.
Saints had a real battle on their hands up front and after conceding a penalty at a scrum, they shipped a second try, with Benetton going to the corner and rumbling forward before the ball went wide for Tavuyara to score.
Allan missed the conversion but his side now held a 15-8 lead with another 20 minutes still to go in the first half.
Things soon got worse for Saints as Biggar was sin-binned for a tip tackle, shaking his head in frustration as he left the field.
Benetton almost scored in the corner but Ahsee Tuala and Cobus Reinach combined to stop Monty Ioane at the last.
It wasn't long before Saints were beaten again though as Rory Hutchinson and Reinach missed tackles, allowing Ignacio Brex to sprint through for a score.
Allan converted to make it 22-8 and Saints were all at sea, looking in real trouble.
Biggar returned from the bin, but Benetton soon added another three points to their ever-increasing tally thanks to another Allan penalty.
Saints desperately needed to cut the 17-point gap before the break and Biggar kicked three penalties to the corner in pursuit of a score.
Eventually it came as Reinach showed his trademark awareness to puncture the Benetton resistance, slamming the ball into the ground after scoring to relieve the first-half frustration.
Biggar converted and Benetton flanker Abraham Steyn saw yellow, meaning Saints would play the first 10 minutes of the second half a man up.
And they came out with real purpose, backed by their travelling army of fans, putting pressure on Benetton.
Francois van Wyk made a big burst through the middle, but Saints couldn't take the chance out wide.
Benetton were starting to struggle in the scrum, with Saints producing two huge shoves against the head, the second of which earned a penalty.
And the black, green and gold soon came close to scoring out wide as Collins was stopped in his tracks by two Benetton players.
Steyn returned to the field without his team having conceded a single point in his absence.
But Biggar was able to slot a penalty for Saints soon after, cutting the gap to seven points with 28 minutes to go.
Saints were on the charge again moments later as Collins grabbed the ball out wide, kicked ahead, gathered and got agonisingly close to the line before being stopped.
Ollie Sleightholme looked set to score soon after, but he knocked on with the line in sight.
However, it wasn't long before Sleightholme made up for it as he skated through amid a backdrop of boos as the home fans felt there was obstruction in the build-up.
Biggar kicked the conversion and it was all square at 25-25 with 23 minutes to go.
Both sides were now just one score from the try bonus point and Sleightholme looked to have a chance to score it for Saints but he was pushed out of play in the Benetton 22.
Reinach, Collins and George Furbank were razor-sharp on the counter and after Furbank did brilliantly to spot space, Collins collected the composed pass and sprinted home.
Biggar added the extras and Saints led 32-25 with 14 minutes to go.
But Benetton soon had a big chance to respond, kicking to the corner and rumbling over through hooker Faiva for their own bonus-point score.
Allan converted and it was all square again at 32-32 with nine minutes to go.
Suddenly, Saints were starting to make mistakes again, allowing Benetton to put the squeeze on, and a high tackle from Lewis Bean gave them a big chance.
But Benetton were unable to take it as Jamie Gibson produced a gargantuan turnover, winning his side a penalty inside their own 22.
Saints pushed forward in search of a decisive moment and it came when Tavuyara was deemed to have deliberately knocked on.
Biggar took his time, settled himself and sent the travelling supporters wild as he slotted the kick that made sure of one of the most memorable European wins in Saints' recent history.
Benetton: Hayward; Tavuyara, Brex, Benvenuti, Ioane; Allan (c), Duvenage (Tebaldi 64); Zani (Traore 53), Faiva, Riccioni (Pasquali 62); Herbst, Ruzza; Pettinelli, Steyn, Halafihi.
Saints: Tuala (Furbank 28); Sleightholme, Proctor (Dingwall 71), Hutchinson, Collins; Biggar, Reinach (J Mitchell 73); van Wyk (Waller 62), Haywood (van Vuuren 77), O Franks; Moon, Ratuniyarawa (Bean 71); Gibson, Ludlam (Eadie 75), Harrison (c).
Referee: Mike Adamson (Scotland)