Liverpool required victory by a two-goal margin to go top of the league for the first time since May 2014 but a set-piece goal against them put that ambition on hold.
An 81st-minute Gareth McAuley goal prevented the Reds from provisionally going top of the board as they defeated West Brom 2-1 on Saturday.
An impressive attacking display provided the win.
Sadio Mané and Philippe Coutinho scored two excellent first-half goals before McAuley reduced the score gap - and Liverpool’s league position.
Klopp praises players
Arsenal had earlier been held to a 0-0 draw by Middlesbrough, while fourth-place Tottenham Hotspur played out a goalless stalemate at Bournemouth.
"I'm really pleased with the performance," Liverpool manager Jurgen Klopp told Sky Sports.
"It was unbelievably difficult against West Brom. You have to always concentrate. They are so dangerous at set-pieces.
"But I think it would have been kind of a joke if we hadn't won today. It was unbelievable, how we played."
Klopp's side have a genuine chance of winning the Premier League this season and five wins in their last six games should pretty much convince most people of that.
Manchester City face Southampton
Pep Guardiola's Manchester City can reclaim first place by avoiding defeat at home to Southampton on Sunday, when Jose Mourinho makes his return to Chelsea with Manchester United.
Klopp made one change to the Liverpool team held 0-0 by United last Monday, with the fit-again Adam Lallana returning to the starting XI at the expense of Daniel Sturridge.
Meanwhile, Tottenham extended their unbeaten start to the season to nine games - something they last achieved in 1990 - but squandered an opportunity to go top in a 0-0 draw at Bournemouth on Saturday.
Sunderland remain rock-bottom after Winston Reid's goal deep into stoppage time earned misfiring West Ham United a 1-0 win at their London Stadium.
Swansea City stayed second from bottom after Bob Bradley's first home game as manager concluded in a 0-0 draw with Watford.