After a goalless first half, the Toffees were two goals ahead within 12 minutes of the second half getting underway courtesy of goals from Michael Keane and Dominic Calvert-Lewin.
That strike was particularly notable for the latter having been heavily linked with a transfer away from Goodison Park throughout the summer transfer window.
However, just as it appeared that Sean Dyche's side were about to respond emphatically to consecutive Premier League defeats without scoring a goal, Bournemouth produced one of the best top-flight comebacks in recent memory.
Antoine Semenyo's goal in the 87th minute could have turned out to be a mere consolation, yet it set up a quite stunning finale where the Cherries scored twice in added-on time.
In the 92nd minute, the diminutive figure of Lewis Cook advanced into the penalty area to meet a floated cross, powering a header into the corner from eight yards out.
At this point, it felt like a point gained for the visitors, but Luis Sinisterra would produce one of Bournemouth's most famous Premier League moments in the 96th minute.
Lurking at the back post, the winger was able to power a header into the net from close range having met an inswinging cross, Everton's supporters left in disbelief after a meltdown that could have ramifications for Sean Dyche.
At a time when Bournemouth now sit in seventh position in the standings, Everton are rock bottom with zero points and a -8 goal difference.
EVERTON VS. BOURNEMOUTH HIGHLIGHTS
Keane vs. Bournemouth (50th min, Everton 1-0 Bournemouth)
Many Everton fans may not like that Keane is continuing to hold down a place in the side, but it was the defender's composed finish that put the Toffees ahead.
An inswinging cross was cushioned down into the path of Keane, who ruthlessly drilled a low half-volley into the bottom corner from 10 yards out.
Calvert-Lewin vs. Bournemouth (57 min, Everton 2-0 Bournemouth)
Everton's second was a touch of the Calvert-Lewin of old, Dwight McNeil supplying an incisive inward pass to the onrushing forward who confidently clipped his shot over the advancing Kepa Arrizabalaga.
Semenyo (87th min, Everton 2-1 Bournemouth)
The first of Bournemouth's three late goals came through Semenyo, who was on hand to tap home from Dango Ouattara' cross.
Cook (92th min, Everton 2-2 Bournemouth)
At the age of 27, Cook had three league goals to his name before this contest, but that has now become four. The defensive-minded midfielder may the kind of run that he is not normally accustomed to, yet he was perfectly placed to power a header past the helpless Jordan Pickford.
Sinisterra (96th min, Everton 2-3 Bournemouth)
Everton were already at a point where they would have been left furious to let in two late goals, but the third will leave everyone associated with the Toffees with much to ponder.
Having gambled to get free at the back post, Sinisterra was found by the perfect inswinging cross, the winger having the relatively simple task in sending a header inside the near post from close range.
MAN OF THE MATCH - LUIS SINISTERRA (BOURNEMOUTH)
Given how the game progressed, we have reason to give the man of the match award to players on different sides, but it is only right that a Bournemouth player receives the accolade.
Sinisterra had been introduced for Evanilson in the 65th minute and ultimately provided the assist for the 92th-minute equaliser and scored the 96th-minute winner.
EVERTON VS. BOURNEMOUTH MATCH STATS
Possession: Everton 47%-53% Bournemouth
Shots: Everton 18-17 Bournemouth
Shots on target: Everton 8-7 Bournemouth
Corners: Everton 8-4 Bournemouth
Fouls: Everton 6-11 Bournemouth
BEST STATS
WHAT NEXT?
After the international break, Everton have the daunting challenge of a visit to Aston Villa when they have now failed to get a single point on the board in their opening three Premier League fixtures.
As for Bournemouth, they play host to an inconsistent Chelsea side later on September 14, already with five points in the bag from three matches.