The hosts raced into a 4-0 lead in the first half with Mirallas - signed from Olympiakos earlier this month - proving the star of the show. The striker netted in the 16th minute, teed up Leon Osman in the 22nd, added his second in the 29th and then squared for Victor Anichebe to finish in the 35th.
Their League One opponents could offer little in response and Everton added another goal after the break, Magaye Gueye volleying home in the 67th minute.
The omens ahead of the match did not look good for Orient, who had managed to beat npower Championship side Charlton on penalties in round one but had lost their opening two league fixtures since then.
The Toffees, on the other hand, were going into the contest off the back of impressive Premier League victories over Manchester United and Aston Villa, and while manager David Moyes had reshuffled his pack considerably by making six changes to his first XI for this game, the Merseysiders did not take long to find their rhythm.
Mirallas was straight into the action with a blocked shot early on and after Gueye had sent a driven effort over and Anichebe headed wide and had a shot pushed away by Orient goalkeeper Jamie Jones, the Belgian opened his Everton goalscoring account with a clinical finish, slotting in from Steven Naismith's through-ball.
The hosts kept the pressure up and within six minutes they had doubled their advantage, Mirallas this time turning provider by flicking Leighton Baines' cross into the path of Osman, who prodded home.
Mirallas had another attempt of his own blocked soon after, but he was not to be denied for long and just before the half-hour mark he made it 3-0, cutting inside in the box and firing in off Jones.
It was proving a torrid night for former Everton trainee Jones, who saved well from Baines in the 32nd minute but should probably have done more to prevent the Toffees' fourth goal from going in three minutes later.
Once again Mirallas was involved, this time delivering a pass which Anichebe sent squirming past Jones from the edge of the area.
The goalkeeper did manage to claw away another Anichebe shot just prior to the half-time whistle, which he and his team-mates will have been relieved to hear.
Moyes made three substitutions at the break, introducing Ross Barkley, Luke Garbutt and Shane Duffy for Osman, Phil Jagielka and Baines. Barkley looked lively in the opening stages of the second half, hitting one shot over and having another charged down.
Orient started to show some signs of life themselves and on the hour, Dean Cox's shot was deflected wide.
That mini-revival did not last long, though, with Gueye cracking in Everton's fifth seven minutes later by connecting sweetly with a cross from Seamus Coleman.
There was a setback for the Toffees soon after as Phil Neville limped off, reducing them to 10 men, but it mattered little in the context of the remaining action, during which Mirallas miscued a shot when presented with the chance to complete his hat-trick.