Despite Redknapp being the overwhelming favourite to become the new Three Lions leader, Bernstein and his colleagues opted for Hodgson.
"I don't want to say too much but I will talk about the process," Bernstein told BBC Radio Five Live. "Four people made this decision. Myself, Sir Trevor Brooking, Adrian Bevington and Alex Hornby.
"We consulted extensively up to 20 people within the game, other managers, players, all sorts of people with a great knowledge and I believe we did a thorough job. We ended up with the right choice, I believe."
Meanwhile, Hodgson is preparing his team to face Montenegro and Poland in their final World Cup qualifiers.