Dunne gave Harlequins a 9-3 lead early on after landing three quick penalties, but Leeds struck back through Duncan Hodge, who put over three of his own to leave the scores level at the break.
The hosts were the first to strike in the second half as winger Phil Christophers broke the stalemate with a try soon after the break.
But Quins hit back with a brace of tries when centre Chris Bell dotted down and then Dunne. The Quins fly-half stretched the lead to 23-16 with the conversions and a try by winger Simon Keogh in injury-time sealed the win for the visitors.
For Leeds:Try: ChristophersCon: HodgePens: Hodge 3
For Harlequins:Tries: Dunne, Keogh, BellCons: Dunne 3Pens: Dunne 3