It was harsh on the Toffees and in particular the brilliant Marouane Fellaini who scored two second-half goals to help his side as they recovered from their early setback and utterly dominate until the final few seconds.
With the spoils set to be heading for Merseyside, the ball was hooked into the box and as defender Seamus Coleman dithered, Dimitar Berbatov's fumble allowed substitute Sidwell to slide in at the back post and steal a point for Martin Jol's men.
It was a breathtaking finish to a very lively contest that had begun so well for the hosts, albeit in slightly fortunate circumstances.
After just seven minutes Phil Neville carelessly chopped down Berbatov on the edge of the box, leaving Bryan Ruiz to curl in a free-kick from his cultured left foot that crept in off the diving Tim Howard and the post.
The goal meant the Toffees had conceded the first goal in each of their last five league games and having rescued at least a point on each previous occasion, they had every reason to attack more adventurously.
Fulham's reluctance to press on merely invited danger and Nikica Jelavic, Coleman and Phil Jagielka all went close with rasping shots.
There was new fire burning in the visitors after the break and having narrowly failed to connect to Jagielka's flicked header, Fellaini grabbed a deserved equaliser.
The Belgian burst through the middle and after the rampant Kevin Mirallas pulled the ball back from the right, the midfielder blasted home (54).
Everton were flying and it was not long before they had the advantage when Jagielka's long ball was chested down on the edge of the box and Fellaini fired in low past Mark Schwarzer after bustling past Aaron Hughes.
The Fulham defence was crumbling and Fellaini had a shot deflected onto the post before Steven Naismith fluffed the follow-up and Schwarzer then palmed away a stunning Fellaini volley.
That save proved to be crucial as Sidwell's late stab earned a last-gasp point for the Cottagers, leaving Everton boss David Moyes distraught on the sidelines and reflecting on a fourth successive draw.
|1||1st Half Goals||0|
|3||Shots on Target||9|
|3||Shots off Target||9|
|35.5||1st Half Poss.||64.5|
|46.7||2nd Half Poss.||53.3|