Hodgson went away from the diamond formation he had used in the warm-up games and returned to a 4-3-3 with Harry Kane starting up front flanked by Raheem Sterling and Adam Lallana.
After a promising but goalless first 45 mintues, a second-half a free-kick from Eric Dier gave England the lead.
However they couldn’t find the second goal and late on CSKA Moscow centre-back Vasili Berezutski equalised for Russia with a towering header.
Tweeting after the game Carragher said there were plenty of positives but added that he felt Leicester striker Vardy should have definitely come on.
Didn't deserve that, played very well. One gripe, if you don't bring Vardy on in that game when else would you? Pace on CA & ageing CB's.
— Jamie Carragher (@Carra23) June 11, 2016
Vardy, who finished as the second top scorer in the Premier League this season with 24 goals and many fans were critical of Hodgson after the game for not bringing him on.
Fellow strikers Daniel Sturridge and Marcus Rashford were also on the bench with Hodgson choosing to keep faith with Kane up front.
England are now second in Group B behind Wales, who defeated Slovakia 2-1 in their opening game, and it is Chris Coleman’s side who are England’s next opponents.