In a contest mostly devoid of offensive quality, Alex Iwobi looked like he would get the last laugh against his former club, only for Beto to nod in the Toffees' leveller deep into second-half injury time and extend the hosts' unbeaten run in the top flight to five matches.
To describe Saturday's evening kickoff as a slow burner would have been an understatement, and murmurs of discontent were growing around Goodison as their side struggled to get out of their own half in the first 20 minutes.
Fulham were not exactly ruthless in the final third, though, and Bernd Leno's net was the first to ripple just past the half-hour mark, but Dominic Calvert-Lewin's strike from an Idrissa Gueye rebound was ruled out for offside.
The second 45 began in a similar vein, with very little to write home about, until two flashes of individual brilliance from Emile Smith Rowe and Iwobi saw the latter break the deadlock against his old team on the hour mark.
A penny for Sean Dyche's thoughts as the Nigerian produced the game's real moment of quality - the Everton boss revealed in midweek that he did not want to let Iwobi go in 2023, but his cash-strapped club could not turn down a £22m package.
Meanwhile, an Iwobi-less Everton side only had one route to goal - punt the ball long to Calvert-Lewin - but the hosts failed time and time again in that endeavour without trying something different.
To Iliman Ndiaye's credit, the 24-year-old was trying his best to engineer an opportunity, but it looked as if it would be a fruitless task against a sturdy Marco Silva defence.
That was until the fourth minute of second-half injury time, when Ashley Young's on-the-money cross found Beto to deny Fulham a fourth-straight win at Goodison Park, and the Toffees now occupy 15th place in the standings while the crestfallen Cottagers lie 10th.
EVERTON VS. FULHAM HIGHLIGHTS
Dominic Calvert-Lewin disallowed goal vs. Fulham (32nd min, Everton 0-0 Fulham)
The Goodison Park crowd have not had much to get excited about so far tonight, and their celebrations are cut short here too!
Gueye sends a stunning 20-yard curler towards goal, which smacks the crossbar and falls kindly for Calvert-Lewin to crash home the rebound, but the offside flag immediately goes up.
From one angle, it looks as if Kenny Tete is playing him on when Gueye lets fly, but an overhead view shows that he is indeed offside, and the on-field decision is upheld by the VAR room.
Alex Iwobi goal vs. Everton (61st min, Everton 0-1 Fulham)
The old Arsenal boys are at it again, and it is an old Everton boy who returns to haunt the timid Toffees!
This one all comes about from a really positive bit of play from Smith Rowe, who drives at the Everton backline and skips past a couple of challenges before laying off Iwobi.
Space opens up for the erstwhile Toffee to pick his spot, and he expertly picks out the bottom corner past three blue shirts. A wry smile from the Nigerian as he deservedly puts the Cottagers in front.
Beto goal vs, Fulham (90+4 min, Everton 1-1 Fulham)
Three of the four 3pm Premier League kickoffs today saw at least one injury-time goal, and Everton have clearly taken some inspiration from this afternoon's events!
Fulham do their utmost to shut up shop late on, but their defensive plan fails, as Young's delicate side-foot cross finds Beto to power a header past a helpless Leno.
MAN OF THE MATCH - ALEX IWOBI
In a game that was crying out for a piece of individual excellence, Iwobi provided one of them, linking excellently with Smith Rowe and managing to get his near-post finish absolutely perfect to beat Jordan Pickford.
As well as his slick strike - which very nearly turned out to be the winner - Iwobi completed three dribbles successfully and caught the eye with his passing range, executing five accurate long balls to no doubt leave Dyche ruing what could have been.
EVERTON VS. FULHAM MATCH STATS
Possession: Everton 40%-60% Fulham
Shots: Everton 10-14 Fulham
Shots on target: Everton 5-3 Fulham
Corners: Everton 4-2 Fulham
Fouls: Everton 8-8 Fulham
BEST STATS
WHAT NEXT?
Neither side are in EFL Cup last-16 action in midweek, giving both managers a full week to dissect the Goodison affair before gameweek 10, which sees Everton head south to face Southampton at 3pm next Saturday.
Meanwhile, Fulham occupy the Monday night football slot in the next Premier League gameweek, as they host London rivals Brentford on November 4.