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.
A cool looking all black rugby cleat with the high performance adidas is known for. Get in the Gear!
Wear the crest of the British and Irish Lions on your t-shirt. A great look for the summer.
$29.99
The All Blacks Performance t-shirt is black with hints of blue from the training jersey. Very Cool.
The New Zealand All Blacks training jersey for 2013/14. Get in the Gear!
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.