A Cristiano Ronaldo hat-trick saw Portugal to a 3-2 win in Stockholm, giving them a 4-2 victory on aggregate and helping them book their place in Brazil.
Zlatan Ibrahimovic's brace had given Sweden hope before his fellow superstar, who scored the only goal of the first leg, sent Portugal through.
Hamren, who looks set to sign a new contract, said his team were hurting after being so close to realising their dream.
"It is a dream and ambition which died and it is hurting everyone and it is really tough – especially when we were close," he said.
"Those two players (Ronaldo and Ibrahimovic) we were talking about before the game are two world-class players and they have shown that in those two games.
"For supporters who are not supporting Sweden it was a good game."
Hamren again played down the contest as being one between Ronaldo and Ibrahimovic despite their influence.
Ronaldo's opening goal was cancelled out by an Ibrahimovic header before a free-kick put Sweden in front.
But the Real Madrid superstar responded with two further goals in what quickly became a two-man show.
"We play as a team. It is always the team playing and I have never said it is Ronaldo versus Zlatan," Hamren said.
"They belong to a team. We have been playing Portugal versus Sweden and they have been stronger than us so you have to congratulate the Portugal team.
"A player like Ronaldo and a player like Zlatan belong to a team so it is the team who has done it."
