The Reds welcomed the Hammers to Anfield having not enjoyed victory over them since January 2015, losing out in all their last three successive meetings in all competitions.
But they got off on the better foot as Adam Lallana spun and fired into the bottom corner in front of the Kop after just five minutes.
The goalscorer gave away a free-kick up the other end just before the half-hour, an opportunity Dmitri Payet needed no invitation to take - curling a free-kick into the bottom corner which Loris Karius received yet more heavy criticism for not keeping out.
Slaven Bilic's side were ahead 12 minutes later when Joel Matip failed to deal with a hopeful Hammers long ball that deflected off the head of Jordan Henderson, allowing Michail Antonio to steal in and poke beyond Karius.
Liverpool struck back early in the second-half, Divock Origi taking advantage of a West Ham goalkeeping error as Darren Randolph dropped a cross at the feet of the Belgian striker just a few yards out.
But they couldn't translate their second-half dominance into goals as they dropped points on home soil for just the second time this season to allow Premier League leaders Chelsea and Arsenal to move further ahead.
Boss says Reds should have seized greater control
Klopp said afterwards that the game was "kind of mixed emotions" because it was "a negative thing" not to win, but they still come away with a point.
He expanded: "Being a goal down in a game against West Ham, with their style-of-play, their situation in the table plus the quality they have, it's really difficult to play against them."
The German said that his side "started brilliantly" against a visiting team that "wanted to defend", meaning they "had to take the ball, play them around and score goals."
He declared it "all good" at 1-0 but suggested the Reds got "a little bit too excited I would say" because they failed to have "a real formation for protection" on the counter-attack as "nearly everyone was involved" in the "offensive situations."
"We had exactly two centre-halves and sometimes Hendo behind the ball," continued Klopp, stating that "all the rest" were further up the pitch which meant it "was a little bit a problem with second balls after."
Yet the Reds boss felt that West Ham's goals did not come "in situations like this" and rather that Liverpool "gave a little bit away in these moments" and "were not completely in control how we could have been."
Klopp: We needed a little bit of luck in the second-half
Reflecting back on the manner in which his side let slip a position of comfort, Klopp added: "West Ham played football, what they're able to do. Then [the] free-kick and the deflected situation with Matip.
"He was fantastic tonight, but slipped a little bit, couldn't control this long ball and Loris was then obviously surprised.
"He was a bit late [coming to meet the ball] and they could score. [It became a] Difficult challenge then."
The manager revealed that in his half-time team talk, they told the players to "cool down" and "ignore the result", instructing them to come out in the second-half with their "usual idea" which would mean they are "able to score one, two or three goals."
Klopp praised the reaction of his Liverpool squad, telling reporters that they "did well" in the second-half and that it "felt a few times" like they spent "the whole 48 minutes in their box", meaning it was "really difficult" because "the whole of West Ham was waiting for us" and they were in need of "a little bit of luck to come through."
The German declared that his side's equaliser - through Origi early in the second-half "was more than deserved" and said his side "tried everything" but "could have done better" as they searched for a winning goal.
He spoke about the need "to find solutions" and insisted Liverpool "did it well" despite accepting that they "crossed maybe a little bit too often" but because of their "occupation of the box."
"Sometimes Divock made a cross and then we have a few players in the box against [Winston] Reid and so. It's not too easy. It's not too simple," continued Klopp, who said that the "second balls were there" as a result of the clearances.
He mentioned what would have been "a fantastic finish" from the hosts' captain Henderson, with his long-distance effort set for the very top corner before Randolph's intervention, which Klopp called "a great save."
Overall, he said that "it's still a point" and that while he might "need a night to feel a little bit better", Liverpool now "have to carry on."
He acknowledged: "Things like this happen. With all goals, we can always do it like surgery and pick at this and that. I completely accept this. But goals like this happen."
Klopp notes "fantastic" elements of performance despite disappointment of setback
The result meant that Liverpool slip six points behind league leaders Chelsea, having been just a point off Antonio Conte's men two games ago.
Klopp said that it is "difficult" for his side to "stay in the race" when Chelsea "win everything" and Liverpool "obviously haven't."
He referred to, when he was asked about being top of the table last month, that they mustn't think about and should rather "collect points as much as possible" because "the season is very long."
Klopp warned that "the most intense time" of the season is "coming now" and said that Chelsea "until now could play nearly the whole time with exactly the same team" and suggested of the other teams that Arsenal were "nearly the same" while Manchester United "had a few problems" and Tottenham Hotspur "lost [Harry] Kane."
But the Reds boss said that his side "are a real skilled team" and that they expect to be "in the top region of the table" which he said is where they "want to be in the decisive part of the season", only then can they "talk about what's possible."
He admitted that being six points behind Chelsea is "not what we wish" but said that they "cannot change it" and can only "do what we can do best."
Again reflecting on their performances agaisnt West Ham, Klopp believed his players showed "a few moments" of "how we are able to play" and called it "fantastic" even with a few of his players - namely Philippe Coutinho and Daniel Sturridge -absent through injury.
The boss insisted that his "offensive options" on the bench were "very young" and that while they "have absolutely no doubt about their quality", this game was "maybe not the best moment" for them.
He noted that Liverpool "will need them in the next few games" and continued: "It's all good in the moment, apart from the results in the last two games. But in both, in large parts of the games, we were the much better team. That's an important thing."
The manager was insistent that he and his team are "still on a very good way" even though they are "not first in the table", insisting "a lot of teams should feel worse and everything is good in this moment."
Reds hopeful Lovren's hamstring pain nothing serious
Klopp was forced to withdraw centre-back Dejan Lovren at half-time, with Ragnar Klavan coming on in his place.
He explained that the decision was because Lovren "had cramp" and that the medical department said it "was in the region" of being "a hamstring issue."
Klopp said that the Croatian "wanted to carry on" but said they "cannot take a risk" in such a situation and hoped that they "took him [off] early enough" to avoid any serious damage.