Sidwell's close-range finish from Dimitar Berbatov's scuffed shot earned a Fulham a scarcely deserved point at Craven Cottage.
Everton fell behind for the fifth straight league match when goalkeeper Tim Howard pushed a seventh-minute free kick from Bryan Ruiz onto the post and into his own net, but recovered well to dominate the game.
Fellaini equalized in the 55th from Kevin Mirallas' cut-back, and the Belgium midfielder put Everton ahead 17 minutes later, chesting down a long ball and shooting past goalkeeper Mark Schwarzer.
Everton moved up a place to fourth, replacing Tottenham after its 1-0 loss to Wigan.
Steve Sidwell rescues 2-2 EPL draw for Fulham vs Everton
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