The men of Gwent last beat Treviso at home in April 2011, with Treviso doing the double over the Dragons last season.
Edwards' side ended a seven-match losing run as their fortunes changed when they saw off Connacht at Rodney Parade, last Friday night.
Centre Pat Leach went over for the only try in the 37th minute and flanker Nic Cudd was named as RaboDirect Pro12 Man of the Match.
Dragons squad to face Treviso:
Tom Prydie, Tonderai Chavhanga, Will Harries, Andy Tuilagi, Pat Leach, Jack Dixon, Ashley Smith, Lewis Robling, Dan Evans, Jonathan Evans, Wayne Evans
Lewis Evans, Nic Cudd, Ieuan Jones, Tom Brown, Jevon Groves, Rob Sidoli, Adam Jones, Ian Nimmo, Dan Way, Nathan Buck, Tim Ryan, Sam Parry, Hugh Gustafson, Steve Jones, Owen Evans, Phil Price, Nathan Williams