City, who won 1-0 at Bournemouth on Saturday, have 71 points while Liverpool are on 70, with both teams having nine games to play.
It promises to be a tense and thrilling finale to the season but Liverpool, who have frittered away a seven-point lead, will need to regain their momentum quickly if they are to push for their first league title since 1990.
"I’m completely fine with chasing,” said Klopp.
"Who wants to be top of the table at the beginning of March? It is nice but there are lots of games to play.”
The result of a typically frantic derby on its own is no cause for alarm for Klopp, but the eight-point swing to City since Jan. 30 is a trend they simply have to end.
There were few chances for either team, with Liverpool’s forward line failing to sparkle and Everton’s impressive industry not matched by craft in the final third.
Everton were desperate to stop their rivals picking up maximum points with manager Marco Silva adjusting his team accordingly as Morgan Schneiderlin was preferred to the more creative Andre Gomes.
It proved a smart move with Liverpool’s Georginio Wijnaldum having limited impact in a crowded midfield.
The home crowd created an intense and passionate atmosphere at Goodison Park, something of a rare occurrence in another disappointing season for Merseyside’s blue half.
Liverpool's Adam Lallana in action with Everton's Seamus Coleman and Kurt Zouma during their Premier League clash at Goodison Park, Liverpool, Britain, March 3, 2019.
"It felt amazing in our stadium. Our fans were amazing and they pushed the team from the first minute,” said Silva.
Mohamed Salah had the two best openings for Liverpool before the break but his soft 15th minute shot barely troubled Jordan Pickford and then the Everton keeper was swift off his line to foil the Egyptian as he raced goalwards.
Everton harried and disrupted Liverpool well but struggled to create opportunities with Theo Walcott unable to capitalise on a rare opening, screwing his shot wide, and Alisson keeping out a Dominic Calvert-Lewin header.
Salah threatened again in the 57th minute as he bore down on Pickford but Everton defender Michael Keane stopped a near certain goal with a perfectly timed last-ditch sliding tackle.
Fabinho, who delivered a masterclass in the modern holding midfielder role, was too slow to react to a nod down from Virgil van Dijk in the box, allowing Everton to scurry the ball clear.
Klopp threw on Roberto Firmino in the 63rd minute but the striker was unable to open up an Everton defence that raised their game to deal a blow to their local rivals’ title hopes.
"I don’t think anyone wants their rivals to win the title,” said the outstanding Keane.
"I think it was a fair result, not too many chances were created by either team.”
Everton fans taunted their rivals at the end, suggesting that Liverpool were not going to win any trophies this season, but Klopp insisted there was no panic in his camp.