In the first of two meetings between the clubs in five days, Bournemouth got the better of two much-changed sides as Jamal Lowe, Junior Stanislas, Emiliano Marcondes and Jaidon Anthony struck on the south coast.
Demarai Gray briefly gave the Toffees hope when he reduced the arrears to 2-1 early in the second half.
But Everton have now won just once in their last seven games with the FA Cup their only hope of ending a 28-year drought without silverware.
Brentford suffered the biggest shock of the night despite taking a third-minute lead through England World Cup hopeful Ivan Toney.
The Gills had just one shot on target but made it count when Mikael Mandron headed in an equaliser 15 minutes from time.
Mikkel Damsgaard is going to Qatar with Denmark, but he was the only man to miss in a high-quality shootout as the visitors won 6-5 on penalties.
Leicester continued their upturn in form with a comfortable 3-0 win over fourth-tier Newport County.
James Justin opened the floodgates just before half-time and Jamie Vardy tripled his tally for the season with a double in the final 20 minutes.
Championship leaders Burnley bounced back from their first defeat in 18 games to Sheffield United on Saturday to beat Crawley 3-1 thanks to two goals from Anass Zaroury.
MK Dons, Lincoln City and Charlton also progressed to the last 16.
Manchester City take on Chelsea in the pick of Wednesday's ties with holders Liverpool, Arsenal and Tottenham also in action.