The Cherries looked dead and buried midway through the second half as Liverpool led 3-1 but three goals in an fantastic final 15 minutes saw them come back to win the game, their first ever victory over Liverpool.
Klopp did admit he was not happy with certain aspects of Liverpool's performance, however - accusing his team of "opening the door for Bournemouth" in the latter stages of a match in which the visitors scored excellent goals through Sadio Mane, Divock Origi and Emre Can.
Jurgen Klopp's side, on a 15-match unbeaten run in all competitions coming into the match at the Vitality Stadium, were 3-1 up with 14 minutes to play but capitulated as substitute Ryan Fraser ran riot for the Cherries.
Bournemouth's Nathan Ake grabbed a stoppage-time victor in a dramatic 4-3 victory over Liverpool on Sunday, as the visitors twice surrendered a two-goal lead in a second-half capitulation.
Liverpool will then return to Merseyside on Wednesday in order to continue preparations for Sunday's Premier League encounter with West Ham United at Anfield.
At Goodison Park, United took the lead in the 42nd minute when Everton goalkeeper Maarten Stekelenburg made an ill-advised decision to come for Anthony Martial's cross and Zlatan Ibrahimovic lofted his shot into the net for his fourth goal in his last three games.
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"We were poor in the first-half but it shows how far we've come". Liverpool were perhaps complacent in their play, as they looked casual after Can's goal and that cost them big time.
"There are players who try to say things in a game, but I never care about this". "If we learn, then it's OK".
Liverpool have only once conceded more than a single goal with the Cameroon global in the side and have conceded just nine goals in 11 games with him, but nine goals in three games without him.
United Manager Jose Mourinho also tried to dwell on the positives after he had sent on Fellaini against eighth-placed Everton. The last goal [Nathan Ake's] was not the most lucky situation for a goalkeeper.
He told Liverpoolfc.com post-match: "We know about our situation, in a good and a bad perspective, so everything is good. I can't change it anymore, so why should I be angry?" he added.