The Villa Park crowd were shocked after only three minutes, when Pienaar fired a shot into the top corner of Shay Given's goal from 25 yards out to give Everton the lead.
The visitors controlled the first period and could have been further ahead when Jelavic found space inside the box, but the Croatian's header sailed wide.
Villa struggled to produce any chances as the home crowd grew increasingly frustrated by Everton's dominance.
The Toffees extended their lead in the 31st minute when Fellaini met Phil Jagielka's cross with a powerful header that Given may feel he could have saved.
Everton continued to control possession and were rewarded once more before the break when Jelavic took advantage of some poor defending to finish well from eight yards out.
The referee's half-time whistle was met with boos by the Villa Park crowd after a disappointing first-half display by Paul Lambert's men.