Griffith, who joins the company's board with immediate effect, has held a number of finance roles since joining Sky in October 1999 during the transition from analogue to digital.
Previously he worked at the investment bank Rothschild, where he advised a range of technology and media clients.
Separately, Danny Rimer has joined the board as an independent non-executive director.
Rimer is a general partner of the venture capital firm Index Ventures, where he has overseen investments in media and internet companies including Skype, Last.fm, Spot Runner, Lovefilm and Joost.
Darroch, BSkyB's chief executive, said: "Andrew is a key member of Sky's leadership team and has made a significant contribution to the growth of our business.
His detailed knowledge of the company and wider industry and his strong technical skills made him the outstanding candidate for CFO.
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