Roberto Martinez's side looked to be coasting to an easy 2-0 win thanks to goals from Seamus Coleman and Steven Naismith, but their defence crumbled in the dying minutes to gift the Gunners a point.
First, Aaron Ramsey bundled home a Santi Cazorla cross and then, with 30 seconds to go, Giroud headed past Tim Howard thanks to some sloppy marking from Sylvain Distin.
It was a cruel blow to Everton, who really should have secured an early blow over their rivals in the hunt for Champions League qualification.
The Toffees were far more clinical and efficient in front of goal right up to the point where Ramsey found the net in the 83rd minute.
In fact, Arsenal looked blunt and impotent in the final third for most of the match.
Alexis Sanchez started as a lone frontman, but, despite his trickery and pace, he never troubled Phil Jagielka and Distin - who have a combined age of 68 - and he was hooked off at half-time.
His replacement Giroud fared just as poorly straight after his introduction, wasting a couple of good chances.
But the France striker came up with the goods in the final minute to send Everton home fans wondering how on earth they did not win.
Source : PA
Source: PA