Karim El Ahmadi scored a consolation effort but Villa owner Randy Lerner needs to produce his cheque book before the transfer window closes to bolster a squad lacking in league experience.
To add to Villa's misery, defender Ciaran Clark was sent off after 58 minutes for a professional foul in what was a first home game to forget for former Norwich boss Lambert.
The Midlands club were carved open at will by the Merseysiders who could have easily doubled their final tally but for poor finishing.
Of equal concern to Lambert will have been Villa's lack of creativity and Everton keeper Tim Howard was not called upon to make a worthy save until his error gifted El Ahmadi his goal.
Everton needed only three minutes to silence the home fans who had given a rousing reception to Lambert with a superb strike from Pienaar.
Steven Naismith, who was making his full debut for the Toffees, laid the ball back into the path of the South African who curled a 20 yarder past Shay Given into the top corner of the net.
Everton threatened every time they ventured forward and a deep cross from Leighton Baines picked out the run of Jelavic at the back post, but he sent his header wide.
The Croatian went close again when his glancing header from a Baines corner was only just too high and then Darron Gibson flashed a 30 yard drive inches over the bar.
Villa started to come more into the game territorially but the lack of a cutting edge which was evident at West Ham last week was again apparent.
Then an error by keeper Shay Given gifted Marouane Fellaini his second goal in two games after 31 minutes to double Everton's lead.
Phil Jagielka floated over a hopeful right-wing cross which the Belgian attacked but his downward header looked to be covered by Given.
But the former Republic of Ireland stopper allowed the ball to wriggle from his grasp and over the line.
Villa were at sixes and sevens and Everton skipper Phil Neville was allowed to advance unopposed into the box but he drilled his shot high and wide from a promising position.
Everton found space at will going forward and it came as no surprise when Jelavic made it 3-0 a minute before half-time.
Baines delivered a left-wing cross and no-one picked up the striker as he drilled a near post shot past Given.
Lambert brought on Brett Holman for Nathan Delfouneso at half-time but play followed the same pattern after the restart.
Pienaar twice carved out opportunities for Naismith and Jelavic and only fierce blocks from Nathan Baker on both occasions prevented Everton increasing their lead.
To make matters worse for Villa, they were reduced to 10 men after 58 minutes.
Clark brought down Jelavic as he bore down on goal and, as the last defender, was sent off for a professional foul.
Given partially atoned for his first half error with a fine save to deny Pienaar.
But he could only stand and watch as Sylvain Distin powered a header against the bar from a Baines corner.
Villa did get score a consolation and it came out of the blue when a swerving long range effort from El Ahmadi deceived Howard who could only parry the ball into the net with 15 minutes remaining.
Villa substitute Andreas Weimann hit a post in time added on but a second goal would have given a flattering look to the final score.