Thomas Partey headed in Smith Rowe's corner for his first goal for the Gunners - having earlier hit the crossbar.
Right at the end of the first half, Arsenal were given a penalty after a long video assistant referee review for a Matt Targett foul on Alexandre Lacazette.
Pierre-Emerick Aubameyang's penalty was saved excellently by former Arsenal keeper Emiliano Martinez, but the Gabon striker put home the rebound from close range.
Villa improved on a thoroughly wretched first half, but Smith Rowe scored Arsenal's third 11 minutes into the second half.
He won the ball in his own half and then finished the move via a deflection and the post.
The visitors got a consolation when Jacob Ramsey curled his first Villa goal into the top corner from 20 yards. They put on good pressure and created a few chances in the closing stages but it was too late.
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