Bernstein leaves after two-and-a-half years in the position.
Dyke, 65, was a director of Manchester United in the 1990s and since 2006 has been non-executive chairman of third tier Brentford, a role he will now relinquish.
"Football has always been a big part of my life whether playing 11-a-side on Sunday mornings or six-a-side on Thursday evenings," he said on the FA website.
"I was brought up in a household where my father was much more interested in whether or not you had won at football than whether you had passed your exams. In my case that was just as well."
"I got involved in how the game was run when I was first involved in buying sports rights as Chairman of ITV Sport in the late 80s and later at the BBC. I learnt a lot in the years when I was on the Board of Manchester United and have seen the other side of the professional game at Brentford," he added.
Dyke said he was a 'big supporter' of financial fair play.
UEFA's financial fair play rules will force top clubs from all leagues across Europe to move towards breakeven or face exclusion from continental competition.
The Premier League agreed new rules last month that mean English top flight clubs could be docked points if they fail to comply with limits on financial losses and curbs on increases in player wages.
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