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Cole Palmer scored a late penalty to make it 4-4 at Stamford Bridge in London.
It was a triumphant return to Spurs for Mauricio Pochettino who got the biggest win of his short Chelsea tenure.
Victory pushed Arsenal up to second place with 24 points from 10 games, two behind Tottenham Hotspur.
Chelsea squandered a two-goal advantage to end up with a point.
The former Chelsea winger joined Real in 2019 as the club's record signing but suffered injuries and struggled to settle at the Santiago Bernabeu during a dismal four-year spell.
West Ham won 3-1 afte Enzo Fernandez missed a penalty to send Chelsea 2-1 up.
The 28-year-old Arrizabalaga will arrive as a temporary replacement for Thibaut Courtois, who will be sidelined for several months after suffering a torn anterior cruciate ligament in his left knee during a training session last Thursday.
Axel Disasi cancelled out Luis Diaz's opener for Liverpool.
Gvardiol is Pep Guardiola's second signing in this transfer window after Croatia midfielder Mateo Kovacic joined from Chelsea.
British media reported the 32-year-old midfielder has signed a contract worth around 86 million pounds ($109.78 million) a year.