Gary Cahill said England had done all they could to get the better of Slovakia in St Etienne.
A 0-0 draw, despite the bulk of the possession, left Roy Hodgson's team second in Group B behind Wales and awaiting a clash with the runners-up of Group F next Monday.
"We tried and tried, time and time again," Cahill told ITV Sport.
"They shut the door in our face. There is disappointment for us tonight. We dominated from start to finish but couldn't shut the door."
Hodgson made six changes but Cahill did not think that was a factor.
"I don't think so," he said.
"We had the ball constantly, we didn't have the cutting edge to open them up. In a game like that you need something special.
"We move on, we know we have a huge game coming up, it gives us time to get back fresh."
England boss Roy Hodgson added: "The fans were magnificent again today. It's a real disappointment for us that we haven't been able to give them a win but I've got to say I don't think we can do so much more than we did.
"We dominated the game from start to finish and had so many attempts and so many chances as well. One day we're going to put those chances away.
"As far as I'm concerned I can't fault the effort and the work that the lads put in and it's a little bit embarrassing that we've played three games now and in each of the three games it's been attack versus defence.
"I actually never thought I'd see an England team being able in a tournament to dominate three games as much as we've done, so that gives me some sort of satisfaction and I'm pretty sure that sooner or later we're going to make somebody pay because if we keep dominating and creating chances like we're doing, we're going to score goals one day."