City twice let the lead slip at Stamford Bridge but looked to have claimed all three points when Rodri put them 4-3 ahead in the 86th minute.
However, Ruben Dias conceded a penalty deep into stoppage time for a foul on Armando Broja. And Chelsea’s Palmer, who came through City’s academy, showed nerves of steel to hammer home the spot-kick.
“We feel exhausted. To concede four goals is not normal for us,” said City midfielder Rodri.
The visitors also had controversial penalty given by VAR to thank for their first goal, with Erling Haaland converting after appearing to initiate contact on Marc Cucurella.
Chelsea showed resilience to fight back three times. Club captain Thiago Silva powered home a header to make it 1-1 before Raheem Sterling netted against his old team from Reece James’ pass.
Haaland slid in his second to restore City’s lead but Nicolas Jackson capitalized on an Ederson error to equalize again. The eighth goal of a classic match was still to come.
“The level of the Chelsea team has risen. It was not our best performance,” admitted Rodri.
The draw leaves defending champions City one point clear of second-placed Liverpool at the Premier League summit.
City face Liverpool in a top of the table clash in two weeks after the international break. And Liverpool warmed up for that showdown with a comfortable 3-0 home win against Brentford.
Mohamed Salah netted his 12th goal of the season as the Egyptian became the first player to score in each of Liverpool’s opening six home league games.
“In that moment, the composure for the first goal is insane,” said Reds manager Jurgen Klopp. “A pretty special player.”
Diogo Jota added a late third as Liverpool maintained their perfect record at Anfield this campaign.
Elsewhere, Aston Villa made it 13 straight home wins in the Premier League by beating Fulham 3-1 to climb into the top five under new boss Unai Emery.
And West Ham eased the pressure on David Moyes with a thrilling late 3-2 victory at home to Nottingham Forest thanks to Tomas Soucek’s last-gasp header.
Brighton’s struggles continued with a 1-1 draw at home to Sheffield United despite leading until the closing stages when the 10-man Seagulls conceded.
The drama-filled Premier League weekend once again highlighted the fine margins between success and failure for the title chasers and European hopefuls this season.