Bernstein, who retires next month, will be replaced by former BBC director-general Greg Dyke with England's place at next year's World Cup in Brazil far from assured.
Bernstein appointed Hodgson to the role and has been a close ally of the former Liverpool and Inter Milan manager even though results have been unspectacular.
The current chairman believes Hodgson will turn around England's fortunes and secure qualification for Brazil.
"He [Dyke] is clearly an experienced, strong-minded individual and he will make his own mind up," Bernstein was quoted as saying. "He will have picked up the vibes clearly.
"I have met with him and passed on that [supportive] view. Roy has been a first-class appointment and I'm very confident he will bring us back to Brazil.
"He brings a lot of attributes to the job: experience, breadth of thinking and interest in the wider development of English football."
England are second in their qualifying group, two points behind Montenegro, who can further extend their lead if they beat Ukraine on Friday.
Critics rounded on the team following a disappointing 1-1 friendly draw with Ireland last week, although the mood was more positive following a 2-2 draw with World Cup 2014 hosts Brazil.
Following the match in Ireland, former England great Gary Lineker tweeted that Hodgson had taken them back to the 'Dark Ages' with his tactical outlook, a comment that did not impress current keeper Joe Hart.
"We're not interested in stuff like that," said Hart. "They can tweet all they want from their sofas - we're out here doing it."
Hodgson's side have won three of their last nine matches, two of them against minnows San Marino.
Three of their next four Group H qualifiers are at Wembley, including a tie with leaders Montenegro, meaning qualification remains in their own hands.
"I would love to be back in Brazil next year but we have a long, hard road ahead of us and we will put everything we can into it," added Hart.
"We have to take the positives though. We beat Spain and Brazil at home in friendlies. No one has taken us apart - we just need to get over the finish line."
