Kai Havertz scored the late winner as big-spending Chelsea fought back for a controversial 2-1 win against West Ham, while Liverpool boss Jurgen Klopp praised Everton keeper Jordan Pickford's "spectacular" display in Saturday's goalless Merseyside derby.
Having embarked on a £250 million ($280 million) spending spree in the summer transfer window, Chelsea are expected to deliver on their new owners' eye-watering investment.
The early returns had been mixed as Chelsea crashed to defeats against Leeds and Southampton.
So when Michail Antonio gave West Ham a shock 62nd minute lead with a close-range finish, the critics were sharpening their knives at Stamford Bridge.
But Ben Chilwell equalised in the 76th minute, just three minutes after the England left-back replaced Marc Cucurella.
Chilwell headed Thiago Silva's lofted ball past his marker, spun around and poked his shot through Lukasz Fabianski's legs from a tight angle.
Kai Havertz cleverly finished off Chilwell's cross to complete Chelsea's escape in the 88th minute of a thrilling finale.
There was still more drama as Maxwel Cornet appeared to equalise in stoppage-time, only for the goal to be controversially disallowed by VAR for Jarrod Bowen's foul on Mendy.
Thomas Tuchel's side remain unconvincing but they are up to fifth place after their third win in six games, while West Ham boss David Moyes was livid with the officials for denying his team a point.
At Goodison Park, there was more VAR intrigue as Conor Coady thought he had put Everton ahead with a close-range finish from Neal Maupay's shot in the second half.
But Coady's goal was correctly disallowed for offside after a VAR check.
There could easily have been goals aplenty but for a man of the match display from Pickford and some fine stops from his Liverpool counterpart Alisson Becker.
"Wow, what can you do? It was a great derby, super intense, with a lot of really spectacular saves," Klopp said.
"We hit the post three times and Pickford made so many saves. I don't know how many derbies I've been in, but it's quite a few and they're always difficult here," he said.
Everton are now without a victory in their first six games this term, while Liverpool's momentum was checked after winning their previous two matches.
Kane reaches milestone
Unbeaten Tottenham made it four wins from six games with a 2-1 victory against Fulham in north London.
Pierre-Emile Hojbjerg broke the deadlock for Tottenham in the 40th minute, sweeping home from Richarlison's pass.
Harry Kane increased Tottenham's lead in the 74th minute from Ryan Sessegnon's cross to become the third highest goal-scorer in the Premier League history, with 188 goals taking him past Andy Cole.
Only Alan Shearer (260) and Wayne Rooney (208) are now above Kane in the scoring chart.
Kane's goal also moved him level with Thierry Henry's tally of 43 in Premier League London derbies.
Fulham's Aleksandar Mitrovic curled home in the 82nd minute, but Tottenham held firm.
Managerless Bournemouth came back from two goals down to win 3-2 in a thriller at Nottingham Forest.
The Cherries, who lost 9-0 at Liverpool last weekend, looked on course for another thrashing as Forest led at the break through Cheikhou Kouyate's header and Brennan Johnson's penalty.
But Philip Billing's long-range blast and Dominic Solanke's acrobatic effort drew them level before Jaidon Anthony sealed the points in the 87th minute.
Ivan Toney scored a hat-trick as Brentford beat Leeds 5-2 at the Community Stadium.
Daniel Podence scored as Wolves beat Southampton 1-0 at Molineux for their first league victory this season.
Podence's simple finish in first-half stoppage time lifted Bruno Lage's side out of the relegation zone.
Newcastle striker Alexander Isak wasted the opportunity to mark his home debut with a goal as Crystal Palace drew 0-0 at St James' Park.
The £60 million ($69 million) man, who scored in his first appearance at Liverpool in midweek, missed two good chances to leave Newcastle without a win since the opening day of the season.
AFP