Everton started well, and were given a golden chance to take the lead when David Jones fouled Aaron Lennon, but Tom Heaton got down low to parry away Ross Barkley's poor effort.
Sean Dyche's side continued to live dangerously though, and they were made to pay when Mirallas fired the hosts ahead midway through the first-half.
Burnley had shown signs of a comeback as the opening half wore on, but matters were made much more difficult for them when Ashley Barnes was dismissed for a second bookable offence moments before the interval.
James McCarthy wasted a great opportunity to put the hosts further ahead after the restart, with Leighton Baines also testing Heaton as Everton continued to dominate.
Burnley frontman Danny Ings also spurned a late chance as his goalless run stretched to eight games – with Leicester City's triumph against Swansea City meaning the Lancashire outfit slumped to 20th place.
In-keeping with their recent good form, Roberto Martinez's men were fast out the blocks, with Barkley and Lennon both looking sharp, and that early pressure should have proved fruitful 10 minutes in when Jones brought down the latter just inside the area.
Despite regular penalty taker Baines being on the pitch, it was Barkley who stepped up, and the midfielder's effort was kept out by Heaton, low to his right.
Lennon fired a warning shot just past Heaton's right-hand upright as Everton continued to press and the Tottenham loanee was at the heart of things as they took the lead, driving through midfield before the ball was worked for Mirallas to prod home at the second time of asking.
Jones blazed over the bar with Burnley's first effort of note just three minutes later, but any Burnley comeback was stunted by a moment of madness by Barnes – already on a booking for a clumsy tackle on McCarthy – as he needlessly slid in on Coleman on the stroke of half-time, earning his marching orders.
The hosts were lucky not to be a man down themselves five minutes after the break, Mirallas escaping with just a booking for a studs-first lunge on George Boyd.
McCarthy should have doubled Everton's advantage just prior to the hour-mark but he drilled into the side-netting after being fed by the vibrant Lennon before Heaton smartly denied Baines' piledriver.
Ings headed over with five minutes remaining as the visitors finally mustered an attempt, but Everton held firm to secure their fourth win in five league fixtures.
