Their goal-kicking scrum half Tito Tebaldi could not convert a 74th minute penalty from a tight angle, and French full-back Julien Laharrague then missed a last-gasp drop goal attempt.
In the end, Treviso, courtesy of Burton's 66th-minute score, held on to take the derby spoils and register their first ever away win in the competition.
Tebaldi and his opposite number Tobie Botes both missed early penalties before Aironi hit the front in an ill-disciplined first half.
Treviso full-back Brendan Williams and Aironi's Giulio Toniolatti and Marco Bortolami all earned yellow cards as the sides fought for every ball.
A Tebaldi penalty was quickly followed by the opening try from flanker Josh Sole, on the half hour mark, as the home side built an 8-6 half-time lead.
Rowland Phillips' side seemed to be within touching distance of their first ever Magners League win when they were awarded a penalty try in the 51st minute, and Tebaldi added the conversion.
But seven minutes later, Treviso were celebrating a converted try from lock Corniel van Zyl and with Botes replaced, his half-back partner Burton assumed the role of match winner in the final quarter.
Last Friday Botes and Burton had kicked Treviso to a hard-fought 15-10 success over Aironi at Stadio di Monigo, and Franco Smith's men were quietly confident of completing the double.
They welcomed back five Italy internationals for this rematch, with Gonzalo Garcia and Tommaso Benvenuti returning in the backs and Ignacio Fernandez-Rouyet, Leonardo Ghiraldini and Manoa Vosawai restored to the starting pack.
The Aironi management opted to make just one change, bringing Gareth Krause in at openside flanker for Simone Favaro.
Treviso threatened in the opening minutes, with Williams and Alberto Sgarbi probing, but Aironi's defence stood strong and they set up an attack from which Tebaldi missed his opening penalty.
In a tight, tactical opening quarter, Aironi had the better of proceedings and the game, played out on a firm pitch, was of much better quality than last week's encounter.
Treviso were able to soak up the pressure and Benvenuti helped set up a kickable chance for Botes whose poor effort went wide.
Tebaldi opened the scoring with a penalty from straight in front of the posts, after Williams was sin-binned. Toniolatti saw yellow in the 24th minute for a dangerous challenge on Botes, and the Treviso scrum half levelled the sides with his first successful penalty.
The game was beginning to open up and Treviso had Aironi stretched when Burton put pace on the ball and winger Ludovico Nitoglia sped through a gap. He needed support but it was slow in coming and the attack petered out.
The hosts hit back and flooded forward after the dangerous Laharrague had found space on the right. He linked with Krause and the supporting Sole, accepting a pass, was able to stretch for the try-line and score. Tebaldi missed the conversion.
Three minutes later, Aironi's poor discipline let them down again when former Italy captain Bortolami was sin-binned. Treviso closed the half with a second penalty from Botes, leaving the game on a knife edge at 8-6.
Aironi made the early running on the resumption. After Bortolami returned to the fray, he passed for out-half James Marshall to dart forward and get within scoring range. Williams tackled him into touch, but the home pack turned up the heat in a set of consecutive scrums and referee Pascal Gauzerre awarded a penalty try, which Tebaldi converted.
Treviso clawed their way back, enjoying more territory and decent possession in and around Aironi's 22. An ill-advised attempt at a counter attack, spearheaded by Laharrague, Tebaldi and Krause, gave Treviso a quick turnover and suddenly they were over for a try.
Smith's charges built their way through a series of phases before a half break created enough space for Botes to offload and set up van Zyl for a seven-pointer.
With a quarter-of-an-hour remaining, Burton nudged Treviso ahead for the first time. Scoring chances were few and far between as time ticked on, but Tebaldi was presented with a 40-metre penalty, out wide, after an offside decision against Treviso.
The 23-year-old's kick had the distance but not the accuracy and despite a huge collective effort in a tense finale, Aironi only had a losing bonus point to show for their 80 minutes of hard graft.
|Aironi Rugby Score Card|
|Benetton Treviso Score Card|
|Cornelius van Zyl||1||5|