Barkley put the Toffees ahead in the 17th minute, finishing at the far post when Brad Guzan turned Gerard Deulofeu's effort his way.
Lukaku then headed home in the 28th minute, before poor Villa defending allowed Barkley to add another three minutes before half-time.
Lukaku notched his second just before the hour, with Barkley and Deulofeu again involved on an afternoon the young trio truly shone.
While Everton boss Roberto Martinez will be delighted to see his team produce another impressive display after the 6-2 thumping of Sunderland last time out at Goodison Park, this was a hugely concerning spectacle for opposite number Garde, whose side remain five points adrift of safety at the foot of the table.
Garde's first game in the job, the 0-0 home draw with Manchester City, had made for an encouraging start, but the challenge facing the Frenchman was laid bare here as Villa's winless run in the league extended to 12 matches - a sequence in which they have lost 10 times.
Everton dominated possession from the off, and it was not long before they broke the deadlock.
A flowing move started with Barkley delightfully flicking a John Stones pass to Lukaku, and the ball then moved slickly on to James McCarthy and out to Deulofeu.
He sent it goalwards and Guzan was only able to parry it into the path of Barkley, who converted with ease.
Ten minutes later Guzan pulled off a fine save to keep out Arouna Kone's shot.
But soon after he was beaten again as Deulofeu tried to play Lukaku in, Micah Richards' block saw the ball break wide to Seamus Coleman and his chipped cross was headed in by the Belgium striker.
It was Lukaku's 10th goal of the season in all competitions and saw him become only the fifth player to reach 50 Premier League goals before turning 23.
It had been a torrid half for Villa's defence but things turned embarrassing before the interval when Everton notched their third.
Lukaku and Barkley helped the ball through and Guzan saved at Kone's feet - but he could not hold on to it, and despite the presence of several Villa defenders, Barkley was able to step in and fire into the net.
Villa had some glimmers of hope after the break, with Tim Howard rushing out to make a decent save when Jordan Veretout tried to pounce on an under-hit back pass by Ramiro Funes Mori, and the goalkeeper then pushing away Kieran Richardson's shot.
But in the 59th minute Everton struck again as their stand-out trio combined once more, Deulofeu feeding Barkley and his tee-up finding Lukaku, who lifted the ball past Guzan.
Lukaku was almost in for his hat-trick towards the end, but Guzan this time kept him at bay.
Source : PA