The visitors from Sixways took control in the first half as they secured a 24-14 lead going into the break.
And John Brain's side then enjoyed a second half shutout as they completed a seven-try win at Roundhay.
Gavin Quinnell led the way with four tries to follow up his scores against Northampton Wanderers seven days earlier.
Wing Marcel Garvey also continued his successful comeback from a fractured eye socket, suffered against Leicester Tigers in the Guinness Premiership in February, with a try.
There were further scores from fly half James Brown and young scrum half Nick Runciman, who celebrated penning a new two-year deal last week by crossing the try line. Shane Drahm kicked two conversions for Warriors.