Leonard landed five penalties and converted a try from flanker Ross Rennie for a personal haul of 17 points at the Stadio Comunale di Monigo.
Leonard booted over three first-half penalties to give the visitors a 9-8 lead at the break after a hard-fought first half.
Leonard fired over his first penalty on 10 minutes and Kristopher Burton levelled the scores with a drop goal for Treviso four minutes later.
Edinburgh led 6-3 courtesy of Leonard's second penalty but Treviso hit back with an unconverted try from Ludwig Nitoglia for an 8-6 lead and Edinburgh were hit by a yellow card for centre Nick de Luca.
Leonard's third penalty edged Edinburgh back in front on the stroke of half-time.
Leonard's fourth penalty in the opening minute of the second half stretched Edinburgh's lead to four points.
Burton landed a penalty for Treviso but Leonard struck again with his fifth to make it 15-11 after 64 minutes.
Nine minutes from time Rennie forced his way over for a try for Edinburgh and Leonard converted from the touchline to give the visitors a match-clinching 22-11 lead.
Gilbert has released a new line of rugby cleats. The Gilbert Virtuo 8S is part of the exciting new product. Check it out.
The entire All Blacks apparel line has been updated for 2013/14. Check out the New Zealand All Blacks polo.
The Nike Tiempo is a solid rugby cleat and one of few styles still made from full-grain natural leather.
The Gilbert Blitz 8S rugby cleat is a great cleat at a great price of $69.99. Get a new pair of cleats today.
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Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
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The USA Rugby Pro Alternate rugby jersey is perfect for any fan of the Eagles. Get yours to wear during the summer Test matches.
The NEW All Blacks 2013/14 jersey has arrived at World Rugby Shop. Dare to wear the colors of the All Blacks.