Manager Roy Hodgson made six changes to the side that dramatically beat Wales in Lens last Thursday, with goalscoring substitutes Jamie Vardy and Daniel Sturridge both chosen to start, while Wayne Rooney and Harry Kane were benched.
The gamble ultimately backfired, however, as England failed to find the breakthrough against Jan Kozak's side despite dominating for long periods.
"In the three games we have had the majority of the possession and here we have dominated play from start to finish and we couldn't unlock the door.
"The rhythm was there, we had the ball constantly but just didn't have the cutting edge to open them up.
"It needs something special in games like that and we nearly did but it just wasn't to be.
"We are disappointed, with the amount of possession we had we should have won the game."
Wales will now face one of the best third-place finishers - potentially an all-British affair against Northern Ireland - on Saturday and England must wait to see who comes second in Group F.
"There is no point harking on about the first game," said Cahill, reflecting on the stoppage time header from Vasili Berezutsky that snatched Russia their only point in the round-robin phase.
"We have to move on, we are second in the group and have a huge game coming up.
"It gives us time to get back fresh. We tried to bring fresh legs in and I thought we looked good. On another day we should win."