Fly-half David Blair converted Talei's try, adding to an earlier penalty, to give the hosts a 10-6 interval lead. In their first competitive outing in Scotland since 1998, Treviso's first half points came from the boot of scrum half Tobie Botes.
Botes quickly closed the gap by landing his third penalty, but Traynor's 48th-minute effort and two more penalties from Blair completed the job for Rob Moffat's men.
Treviso were the first ever Italian club to visit Murrayfield. They made the trip on the back of their fourth home victory of the league campaign, a 24-17 defeat of Connacht.
Italian lock Gonzalo Padrò was brought in to make his competitive debut for Franco Smith's side, with hooker Franco Sbaraglini captaining the visitors in the absence of Leonado Ghiraldini.
Tim Visser, the league's joint top try scorer this season, pulled up with a groin strain in the warm-up, and flanker Alan MacDonald fell victim to a stomach bug, prompting the Edinburgh management to make two late changes. Lee Jones and Scott Newlands were promoted into the line-up.
In wet conditions, Blair and Botes swapped early penalties and Treviso, despite missing a number of their internationals, showed some early fire.
Their pack made a solid start and a lively break from new man Padrò brought the visitors into the 22, with out-half Chris Burton's subsequent chip almost leading to a try. The ball went out of play after winger Benjamin de Jager failed to grasp it and the head of scrum half Greig Laidlaw came to Edinburgh's rescue.
Edinburgh threaten from long range, using turnover ball to attack at pace and out wide, but Treviso pushed ahead in the 28th minute when Botes' second penalty success rewarded his forwards for a determined spell of phases.
Edinburgh hit back four minutes before the break. The bustling Talei shrugged off three tacklers on the way to the try-line, after man-of-the-match John Houston had straightened and found a gap near the halfway line.
Blair added the conversion for a 10-6 scoreline, before some clever kicking from Burton and an ill-advised quick lineout from Jones and Mark Robertson gave Treviso a late chance to build a score. Edinburgh's defence held firm and the four-point margin remained fo the second half.
Talei's third try in as many games at Murrayfield should have inspired more of the same from Edinburgh, and they did ease the tension somewhat with their second try, scored from a close range ruck by Traynor.
It was a one-point game when Botes punished Scott Macleod for interference in the lineout. Thankfully for the majority of the 2,422 crowd present, they soon wrestled back some control.
Botes missed a penalty, but a deft kick and chase from Jones got the home side into position and after full-back Jim Thompson had gained further yardage, Traynor picked off the base of a ruck and plunged over in the left corner. Blair's missed conversion left it at 15-9.
Missing the influence of some of their internationals, Edinburgh again failed to kick on amid a series of disappointing errors. Gradually, as the half progressed, the Scots looked more potent and it was surprise that they could not add to their try count in the end.
With Treviso now forced back on the defensive, Edinburgh sensed more scores. A thunderous break from replacement Stuart McInally should have amount to more than a turnover for the Italians.
Blair's second successful penalty of the night - from in front of the posts - left nine points between the sides, with the worsening conditions making it difficult for both sets of players to construct attacks.
The 25-year-old Blair tagged on another penalty to effectively put Edinburgh out of sight, and while they deserved the victory, Moffat's charges will be annoyed that their performance was not what it could have been - especially given their improved play in the third quarter.
|Edinburgh Rugby Score Card|
|Benetton Treviso Score Card|