England looked on track for victory in their opening Euro 2016 game after Dier's sublime free-kick put Roy Hodgson's side in front, but Berezutski sent a looping header beyong Joe Hart in injury time.
Despite the disappointment of dropping two points in the closing minutes, a disappointed Dier praised his team's display prior to the late sucker punch.
"I thought the level of performance was extremely high," Dier told ITV, agreeing it was the best showing he has been involved in at inernational level.
"I thought we did everything right. We controlled the game, we played it out from the back, we pressed when we needed to.
"We won the ball back, we countered, I think it had it all from us apart from controlling it at the end.
"Of course [it feels like a defeat]. We're very disappointed because I think we played very well all game and to give away the draw at the end, it's very disappointing."
Dier denied that a lack of experience was the reason for Russia's late equaliser as Berezutski towered over Danny Rose in the penalty area, insisting that he and his team-mates will pick themselves up and go again.
"It's not inexperience, it's not got anything to do with experience. We just didn't see it out in the right way and it's disappointing but we didn't lose," the Tottenham man said.
"We've got to pick ourselves up and go again. I think we can take a lot of positives from the game but it's just disappointing we couldn't see it out.
"I just like to talk game by game and we've got another big game - every game is a big game - and we're looking forward now to the next game. We've got to put this behind us - it's gone."
England's next outing is a clash against Wales, who beat Slovakia in their Group B opener, on Thursday.