The Toffees responded with two chances of their own in quick succession, but on both occasions Bryan Oviedo put his shot narrowly past the post.
The visitors continued to push for the opener and they almost got it in the 17th minute when Mirallas clipped the ball back to James McCarthy, who volleyed his effort into the arms of Ben Foster.
Chris Brunt then made a fine block at the far post to deny Oviedo a certain goal, before the hosts went up the other end and almost scored themselves, Zoltan Gera drawing a good save from Howard after being played in by Nicolas Anelka.
The deadlock was finally broken with just four minutes of normal time left in the half when Sylvain Distin's long ball was flicked on by Romelu Lukaku to Mirallas, who calmly slid it past Foster.
If Everton can hold on to their lead for the remainder of the match, then they will move into the top four of the Premier League table.