The midfielder, who was restored to the starting lineup for today's game, did put his side 1-0 up five minutes later, though.
Osman sold a dummy to a Fulham defender before curling a finish into the far corner with his left foot.
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Gerard Deulofeu, the second Everton change, almost doubled his side's tally with a fizzing cross-cum-shot that flew just wide of the far post, and the 20-year-old went close once more in the 28th minute.
Osman came close to doubling both his and Everton's advantage 10 minutes before the break, but his effort flew past the post after Stekelenburg had thwarted Ross Barkley with a good save.
A minute later, Giorgos Karagounis had Fulham's best chance of the half, but the veteran midfielder lashed his effort wide from a tight angle.