The Three Lions have enjoyed the lion's share of possession and chances in all three of their group games, but have only managed to win one of those.
They were forced to settle for a second-place finish in Group B as a result of a "frustrating" goalless draw against Slovakia this evening, but Hodgson, who made six changes for the contest, insists that they will take their chances sooner rather than later.
"It is [frustrating] when you dominate three games and only manage to win one. The fans were magnificent and it is disappointing for us not to be able to give them a win. We can't do much more, we dominated the game from start to finish, we had so many chances and one day we will put them away, I can't fault the effort and work of the players," he told ITV Sport.
"It is a little bit embarrassing, it has been attack vs. defence in all three games and I never thought I'd see England dominate three games like we have done. Soon we will make someone pay, we will score goals one day. If we had won the game people would say we didn't miss [the players who were left out] and when we don't they say the team selection is wrong, I am used to that one.
"We brought players on which was useful. It kept up our domination but I am not certain it would have made a vast difference because we would still playing against 11 players in defence. Finishing second is a disappointment but we are still in the last 16 and who is to say the team we will play will be that much stronger? You just don't know, the way we are playing I am not frightened of anybody."
England will face the runners-up of Group F in Nice in the last 16 on Monday, June 27.