Everton striker Dominic Calvert-Lewin is controversially sent off but the 10-man Toffees hold on to force an FA Cup third-round replay with Crystal Palace.
There was no stopping Manchester City's second-half steamroller at Goodison Park after Everton had taken the lead through Harrison in a strong first-half display.
Tottenham Hotspur move into the Premier League top four after two early strikes are enough to beat Everton, with Richarlison scoring against his former club.
Tosin Adarabioyo is the hero, converting the winning spot-kick in the shootout, as Fulham beat Everton on penalties to reach the semi-finals of the Carabao Cup.