The Premier League side were always in control and went ahead after 16 minutes when Kevin Mirallas slammed home from deep inside the visitors' penalty area.
It became 2-0 to Everton six minutes later when Leon Osman found the target from close range, and Mirallas added his second goal of the evening in the 29th minute.
Everton went into the half-time interval 4-0 up following Victor Anichebe's goal from outside the penalty box in the 35th minute.
David Moyes's side continued to dominate proceedings at the Goodison after the restart, with Magaye Gueye netting the Toffees' fifth goal from the edge of Leyton's 18-yard box.
Everton had chances to increase their tally during the remainder of the game, but eventually had to settle for a 5-0 victory.