The Belgian marks his comeback from suspension with the equaliser but David Moyes' side will be disappointed not to claim all three points over a poor Gunners team
The visitors made the perfect start as Theo Walcott put the Gunners ahead with less than a minute on the clock, bending home after an Aaron Ramsey pass.
But Fellaini pounced on one of many sloppy Arsenal passes to drive past Wojciech Szczesny and earn Everton a share of the spoils.
David Moyes made three changes for the visit of the Gunners, with the Belgian midfielder returning from suspension alongside Darron Gibson and Tony Hibbert, replacing Brian Oviedo, Thomas Hitzlsperger and Johnny Heitinga.
Arsene Wenger also took the opportunity to rotate his squad. In came Walcott, Jack Wilshere, Bacary Sagna and Thomas Vermaelen, with Alex Oxlade-Chamberlain, Carl Jenkinson and Kieran Gibbs dropping to the bench. Lukas Podolski missed out altogether.
And it was the Gunners who made a rapid start to proceedings. With less a minute on the watch, Walcott collected a slide-rule pass from Ramsey on the left side of the penalty area, before bending it into the top corner via a slight deflection.
The early goal seemed to settle the home side after their shaky opening and following the withdrawal of Laurent Koscielny to injury, which saw him replaced by Gibbs, Arsenal retreated further into their own half as the 45 minutes wore on, giving the ball away regularly.
They were eventually punished for their carelessness when Fellaini intercepted a dire pass from Sagna before sliding a 20-yard shot beyond Szczesny for the equaliser.
Jelavic should have completed the first-half turnaround after he nipped round Per Mertesacker in the box, but the striker could only blaze a right-footed drive wastefully over.
The Gunners were producing a toothless display, riddled with soft errors and disjointed attacking play, with Everton in stark contrast – and especially Jelavic and Fellaini - looking dangerous in the final third.
And as the half-time whistle drew near, the Croatian bore the brunt of Baines’ frustration when the England left-back burst clear only for Jelavic to nip it off his toes in an offside position.
The second period began in a similar vein, with Steven Naismith only denied by a last-gasp Gibbs challenge and Sylvain Distin’s header well saved by Szczesny as Everton began to turn the screw.
A committed home display was producing everything but a goal although Steven Pienaar was unfortunate to see a vociferous penalty appeal turned down after a clumsy challenge from Mikel Arteta.
Arsenal nearly secured the breakthrough the second half desperately needed but Tim Howard was alert enough to paw behind an innocuous Ramsey through-ball.
The Gunners though were continuing to be outmuscled, outharried and ultimately outplayed as both sides pushed for a winner but were producing the better chances - substitute Gervinho was inches away from converting an Olivier Giroud cross with the goal gaping.
And despite a frantic end to proceedings, both sides had to settle for the draw after an enthralling clash.