Wayne Rooney gave England an ideal start with a fourth-minute penalty in Nice, only for Ragnar Sigurdsson and Kolbeinn Sigthorsson to turn the match on its head before the 20th minute.
England struggled to break down their opponents throughout the remainder of the game, becoming more panicked in their play, as Iceland held on for a place in the quarter-finals against France as their remarkable campaign goes on.
Although understandably elated at his side's progress, joint head coach Lagerback still had words of praise for his opposite number.
"One of the reasons I sit here is Roy Hodgson. Huge thanks to him," he said in his post-match media conference.
"He changed the coaching ways in Sweden and I have to thank him for that. I have a lot of respect for him."
Lagerback went on to hail his players who, along with the nation's supporters, have captured the imagination of many in France.
"This was a team performance. We are well organised, everybody is doing his job. Everything is in place on the pitch. We deserved our win," he said.
"Football is about winning, about not conceding and scoring yourself. This team showed what it can do. The players were absolutely great.
"We tried to be a bit cooler in possession. We are taking small steps. But we must keep improving. I have a lot of respect for the players when they take less risk when we are defending a result.
"You don't feel like this on a lot of occasions, especially with a team like Iceland. There is a lot of pressure and things going on when you take on a team like England. There was a huge happiness after the final whistle.
"The France game will be even bigger. We are playing the hosts in a quarter-final in Paris. That's even bigger than then England game. We want to improve a little bit.
"This match will have an influence on the players. Some of them deserve contracts at bigger clubs than they are at now. Plus this is a huge experience to go through something like this. These players can still develop and do even better. The players must keep their feet on the ground, though."