Liverpool nearly grabbed all three points when Jordan Henderson hit the crossbar in injury time with a long-distance shot, but even Klopp agreed that a win would have been undeserved.
"The start of the game was really bad from us," Klopp said. "No direction, no positivity, no opening up, no breaking lines, nothing (of) what you have to do."
It was a different story in north London, where Tottenham reacted superbly to going behind in the 12th minute.
Ryan Sessegnon and Eric Dier put Spurs in front by halftime and Southampton defender Mohammed Salibu netted a comical own goal when trying to clear the ball off the line in the 61st minute. Dejan Kulusevski capped a stellar performance by slotting in the fourth two minutes later.
Newcastle beat newcomer Nottingham Forest 2-0 after Fabian Schar finally broke the deadlock with a shot into the top corner from just outside the area in the 58th, and Callum Wilson added the second.
There was a distinct American flavor to Leeds' 2-1 win over Wolverhampton at home.
U.S international Brenden Aaronson played a key part in the winning goal as he rushed in to meet a cross from Patrick Bamford, although it was credited as an own goal by defender Rayan Ait-Nouri, who got a touch on the ball.
Aaronson and fellow U.S. international Tyler Adams made their first start for Leeds, which is managed by American coach Jesse Marsch.
Promoted Bournemouth beat Aston Villa 2-0 thanks to goals from Jefferson Lerma and Kieffer Moore.