The England full-back, who netted on international duty last weekend, got himself on the scoresheet again when he applied the finish to a fine Toffees move.
The two players had earlier combined to set Pienaar up for a first-time strike, but he blazed wide from Baines's cutback, while he also sent a great opportunity off target as the half-time whistle approached.
Baines was also influential at the other end and capped a superb first-half display by clearing a Vurnon Anita shot off the line.
He twice tested replacement keeper Steve Harper with powerful low efforts after earlier going close with a curling long-range strike.
A fantastic 45 minutes was soured slightly for Everton when an injured Jelavic was forced off and replaced by Victor Anichebe.