Sweden's Karlsson finished fourth in Paris, two strokes behind winner Graeme McDowell. Northern Ireland's Hoey was seventh, four shots back, and Frenchman Riu was eighth, five shots off the pace. British Open qualifying spots were available to anyone who finished in the top 10 and was not already eligible.
The British Open will be held July 17-20 at Royal Liverpool, Hoylake.
Karlsson, 44, has played in 16 British Opens, with a best result of T-5 in 1992 at Muirfield. Hoey, 35, has made just two British Open appearances, missing the cut in 2001 and 2012, both times at Royal Lytham & St Annes. Riu, 29, will be making his first start in a major championship.