They failed to clear the ball and Scott Arfield produced a superb half-volley for his first goal in 11 months.
And while that lead lasted only a few minutes - Mohamed Salah making it 1-1 at half-time - the Clarets defended superbly to stop Liverpool breaking through again before nearly pinching a famous win themselves through inspirational stand-in skipper Ben Mee in a breathless finish.
Tom Heaton underwent surgery on an injured shoulder earlier this week and was replaced in goal by Nick Pope.
"That's a very important thing, so we have quite a good form or shape".
Mohamed Salah was quick to restore parity for the Reds after Scott Arfield gave the visitors an unlikely lead at Anfield inside the first half an hour, but despite their domination of the contest from there on - with some 35 shots in all - Klopp's side were unable to find a victor.
Liverpool's Alex Oxlade-Chamberlain in action against Burnley in Anfiled, September 16, 2017. Daniel Sturridge replaces Sadio Mane in the front three, with the Senegal worldwide serving the first of his three-match ban. James Tarkowski directed a long diagonal towards Robbie Brady, who got the better of Trent Alexander-Arnold in the air. Emre Can wasn't alert to his run from midfield and Andy Robertson had realised the danger too late - it was the kind of collective defensive failure the Reds boss had referenced pre-match.
Was Luke Rockhold Calling Out Michael Bisping? Champ Isn't Sure
Rockhold had both hooks in and worked to flatten Branch out while still landing bombs, before long, Branch tapped to the strikes. With that, Luke Rockhold was back in the win column and back in the title hunt. "I wanted to take my time and be patient".
With Burnley managing only four shots in total at Palace's goal, half of them being off-target, a lack of creativity afflicted the Burnley performance. "How I've said, he has trained outstanding in the few sessions we've had so far together since he was back", added Klopp. In the last five matches, Liverpool has won three matches, lost one match and the other match ended in a draw.
Liverpool's defence craves confidence and with no new blood in the dressing room, consistency looks to be an obvious route to providing it.
Today's match ended as one of the typically disappointing Liverpool games fans have come to nearly expect against smaller sides such as Burnley.
"It feels odd, it feels wrong but it is a fact we drew", Klopp said in his post-match press conference.