The Lions had to come from behind to win, having slipped 18-3 behind early on after tries from Wilhelm Koch and Rayno Barnes.
Tommy Bowe's try for the tourists cut the deficit, and although Bees Roux crossed for the Royal XV, further tries from Lee Byrne, Alun Wyn Jones and Ronan O'Gara ensured an opening win.
Meanwhile, England were beaten 33-26 by an all-star Barbarians side at Twickenham.
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