They did eventually break the deadlock 10 minutes later when a nice passing move resulted in Seamus Coleman pulling the ball back for Pienaar, who slammed his first-time shot past Schwarzer.
The Toffees should have doubled the lead 10 minutes before the break when another good move released Fellaini, but the Belgian put his shot wide when he should have done better.
Fellaini had another good chance soon afterwards, but this time he was denied by Schwarzer having latched on to a Leighton Baines cross.
Mladen Petric had Fulham's first good chance just two minutes before the break when he tried to get on the end of Kieran Richardson's ball across the face of goal, but the Fulham striker just couldn't make contact for what would have been an easy tap-in.
The visitors had another good chance 10 minutes after the restart, but Richardson had acres of space on the edge of the box, but he blazed his shot over.
Fellaini then completed his disappointing day in front of goal by missing a sitter with five minutes left. Schwarzer failed to collect a Baines cross, but the Belgian couldn't turn the ball in at the back post.