The Cherries have recently lost their way in the top flight, suffering defeats from winning positions against each of Tottenham Hotspur and Leeds United, but Gary O'Neil's much-changed lineup stunned the Toffees at the Vitality Stadium.
Jamal Lowe and Junior Stanislas put the Cherries ahead before Demarai Gray halved the deficit with 23 minutes remaining, but Emiliano Marcondes and Jaidon Anthony scored during the latter stages to send Everton out of the competition.
Leicester City enjoyed a more routine evening as a James Justin goal just before half time and a Jamie Vardy double earned the Foxes a 3-0 triumph over Newport County.
Meanwhile, Brentford have bowed out of the competition after losing on penalties to Gillingham, Mikkel Damsgaard hitting the crossbar from 12 yards after 11 successive conversions.
Ivan Toney had initially given the Bees an early lead, but Mikael Mandron equalised for the League Two outfit soon after being introduced as a substitute.
Another shock came at Ashton Gate as Lincoln City, who were defeated by non-league Chippenham Town in the FA Cup on Saturday, posted a 3-1 success at Championship side Bristol City.
Matty Virtue, Ben House and Paulie O'Connor gave the Imps a 3-0 lead after just 49 minutes, with Tommy Conway's effort 10 minutes from time proving to be no more than a consolation.
Burnley came under threat of a surprise defeat at home to Crawley Town, but the Clarets came from behind to win 3-1 courtesy of goals from Ashley Barnes and a late Anass Zaroury double.
MK Dons came through their all-League One encounter with Morecambe by prevailing by a 2-0 scoreline, with Warren O'Hora and Matthew Dennis getting on the scoresheet.
Lastly, Charlton Athletic booked their place in the last 16 with a penalty-shootout win at Stevenage after finishing level at 1-1 at full time.