Norway's dream run came to an end after Andreas Schjelderup gave them the lead before Jude Bellingham equalised before half-time and then scored the winner early in extra time to send England into the semi-finals.
Despite the disappointment, Solbakken said he could not have been prouder of his players after they reached the last eight and stunned Brazil in the Round of 16.
"We have been playing at the highest level, and the margins were not in our favour, but that's life, and now we need to breathe a bit," Solbakken said.
Norway returned to the World Cup for the first time since 1998 and quickly became one of the tournament's biggest success stories. Led by Erling Haaland, they impressed throughout the group stage and won over supporters with their exciting football and the now-famous rowing celebration performed by their fans.
Solbakken said his players had proved they belonged among the world's best teams.
"This is the way it is, this is at the top level, it can't get any higher than this and we had a few problems in the beginning, the first 20 minutes, but I really applaud the boys," he said.
The team's impressive run sparked huge excitement back home, with demand for Norway's red home shirts soaring as fans packed stadiums and public viewing areas to support the national side.
Solbakken thanked the Norwegian supporters for standing behind the team throughout the tournament.
"I hope that the summer of '26 has been okay for everybody and that we've done something good together. We have had fantastic support from all of Norway, and we lived up to the hype, and that's what I'm really proud of," he said.
"We said that we would step up when we got to the World Cup from the fantastic qualifiers, and we've managed to do so, and the boys did so. And we have been here together for six-and-a-half weeks. I don't see that people have been bored for a second, and that's good to know."
Norway also had a goal from a set-piece ruled out after Haaland was judged to have pushed a defender, while there was speculation that the ball may have struck an overhead camera wire before England's winning goal.
However, Solbakken refused to blame the officials or search for excuses after the defeat.
"There are many situations today. And, well, maybe we have been looking at many of them in the World Cup, so these things happen. We just want to wish England the best of luck," he said.
