Having seen both Manchester City and Arsenal, who headed into the weekend ahead of them on goal difference, score four goals in victories earlier in the day - the Reds followed suit with an impressive showing in the capital.
A breathtaking, and frantic, first-half saw Emre Can open the scoring after 16 minutes, only for Dejan Lovren's error to gift James McArthur a header to make it 1-1 two minutes later.
But centre-back Lovren rectified his mistake almost immediately up the opposite end of the pitch, heading in to re-establish the Reds' lead inside a short spell.
Yet just as Liverpool looked comfortably on top, Palace equalised a second time - again through McArthur as he powered another header in from Wilfried Zaha's cross.
The visitors had time to make it 3-2 before the break, a fourth headed goal of the game as summer signing Joel Matip leapt highest for his first goal for the club.
But the game was finally put out of the Eagles' reach by Roberto Firmino, who cutely lobbed the on-rushing Steve Mandanda after meeting Jordan Henderson's terrific through ball mid-way through the second-half.
Liverpool saw it out from there to stretch their unbeaten run in all competitions to 11 games, having won nine of those, with only goal difference separating them from the summit of the league.
Two changes for Palace and 10 for Liverpool
Alan Pardew made two changes from the hosts' defeat to Leicester City last weekend as captain Scott Dann made his first start in over a month, replacing Damien Delaney on his return from injury.
In midfield, Chung-yong Lee came in for Andros Townsend while Wilfried Zaha made his 200th league appearance for the club and Christian Benteke led the line against the club he left for £32 million in the summer.
With James Milner having missed training earlier in the week, Alberto Moreno replaced him at left-back for Liverpool in his first Premier League appearance since the opening day.
Emre Can was preferred in midfield over Georginio Wijnaldum as Klopp made 10 changes from the mid-week EFL Cup win over Tottenham Hotspur, but only one change from their last line-up in the league.
Sensational six-minute spell sees Lovren go from zero to hero
It was the visitors who started off the better side and though they created few chances in the early stages, it was no surprise when they broke the deadlock after 16 minutes.
Following a lengthy spell of possession, Moreno shared a one-two with Coutinho down the left-hand side and his first-time pass from the return ball fell perfectly for Can - who ran on and stabbed an effort into the bottom corner courtesy of a deflection off of Dann.
Sadio Mane almost had the opportunity to double their lead within a couple of minutes, but Dann was slightly quicker to a through ball, managing to poke it beyond the Senegalese winger and into the safety of Mandanda.
And immediately up the other end from the goal kick, Liverpool shot themselves in the foot as they gifted Palace an avoidable equaliser. Lovren got his pass back intended for Karius all wrong as it looped up into the air, allowing McArthur to steal in and win the loose ball, heading over the goalkeeper and into an empty net.
But within an incredible six-minute spell it was 2-1, Lovren making amends for his error by rising beyond Dann and heading down between the legs of Mandanda from Coutinho's corner.
From there, Liverpool re-asserted their control of proceedings and looked the more likely team to score again. The lively Moreno reached the by-line to send a deep cross in towards Firmino who was narrowly unable to reach the delivery.
Moreno soon hit the post as they controlled open play. The Spaniard made an overlapping run down the outside of Coutinho, who slipped it into the left-back's path - but his fierce strike rocketed off the outside of the post from an angle.
The Reds failed to take advantage of their dominance and were made to pay for it after the half-hour, Moreno standing too far off Zaha and his cross found McArthur, who guided his header inside the near post for his and Palace's second of the evening.
Matip restores Reds' advantage before the break
In what was a breathless first-half, chances came and went - mainly for the visitors - as Mandanda made a stunning save to deny Coutinho, tipping his header from six yards out on to the post and away from danger.
Replays showed Liverpool could have been awarded a penalty when Joel Ward's arm seemed to strike the ball in mid-air, although referee Andre Marriner was unmoved.
Liverpool kept possession and soon fashioned another opportunity which Mane was guilty of wasting, somehow firing over the crossbar from just 10-yards when it seemed easier to score.
Klopp's charges could and should have been out of sight, Coutinho picking out Lallana in space to his left but the midfielder looked to find his teammate in space at the back post rather than shoot and Palace watched it out of play.
But Liverpool once again took the lead shortly before the break, with a pulsating opening 45 minutes continuing to excite.
Defender Matip found five yards of space at a corner to unleash an unstoppable header past Mandanda for his first goal for the club, making it 3-2 going into the break.
That meant it was the first Premier League game since 1998 in which four headed goals were scored in the first-half alone.
Half-time provided a welcome pause for thought following a five-goal first-half, the most of any Premier League game so far this season.
And the opening stages of the second period were tame - at least compared to the first-half - until Karius was required to dive low and parry Benteke's drive on the turn wide at his near post.
Liverpool should have had a fourth when he raced through on goal and on to Coutinho's sliding through ball, but his attempt to find the far bottom corner saw Mandanda save well with an outstretched leg.
But Palace exerted more threat from open play in the second-half, Benteke's headed lay-off finding Yohan Cabaye before the midfielder returned the ball to Benteke in space in the box, but his attempt curled straight into Karius' grasp.
Firmino puts the game out of reach
Palace were appealing for two penalties in quick succession on the hour, Zaha seemingly tripped Lovren before Matip went in late on Benteke moments later - though Marriner gave nothing for either incident.
An increasingly frustrated Palace felt they warranted a penalty again in the 69th minute when a coming together between Can and Zaha saw the latter fall to the floor demanding a spot-kick, which Marriner again waved away.
And their frustrations were compounded with 20 minutes remaining as Henderson's incisive defence-splitting deep through ball invited Firmino to lift the ball over Mandanda and into the empty net beyond him to move his side on to the same points as City and Arsenal in third.