The holders won the second leg of the last 16 tie 1-0 in the Andalusian capital but a 2-1 defeat in last week's home leg meant they were eliminated on the away goals rule, Guardiola's first taste of failure since he took over last season.
"We are out and it hurts to be eliminated from this competition," the 38-year-old said at a news conference.
"It's not good news, even if people will say that we can now concentrate on La Liga and the Champions League," he added.
"In sport we are not taught how to lose and this has affected us."
Barca, who can no longer match last year's feat of winning an unprecedented six titles, were last eliminated from a competition in April 2008 when they lost to Manchester United in the Champions League semi-finals.
Guardiola had come under fire for using a weakened side in the first leg but he fielded all his stars for the return, including Lionel Messi, Zlatan Ibrahimovic, Xavi and Andres Iniesta, and Barca came agonisingly close to going through.
Sevilla captain and goalkeeper Andres Palop was in stunning form, producing at least four brilliant saves, and Messi was denied by a post midway through the second half.
Guardiola blamed himself for the defeat but said he had been pleased at the players' disappointment as it showed they were still hungry for success.
"With this squad and these players maybe I should have got more out of them in the Cup and not lost in the last 16 and I am sorry that I failed," he said.
"It's a good sign that the dressing room is affected, that it affects them to lose. I am happy about that," he addeed.
"I saw reactions in the dressing room that pleased me. This shows we still have the hunger to win.
"We went down staying faithful to our style of play and now we need to get ourselves together again for the future."
Barca's next match is also against Sevilla, a La Liga clash at the Nou Camp on Saturday.
