The Gunners midfielder put the home side ahead on six minutes before Laurent Koscielny stooped to head home the second eight minutes later.
Ramsey made the score 3-0 with a deflected shot from the edge of the area while January signing Pierre-Emerick Aubameyang, making his Arsenal debut, chipped over Jordan Pickford on 37 minutes.
Everton, 4-0 down at half-time for the first time in the Premier League, responded after the break and after Oumar Niasse hit the post, they pulled a goal back through Dominic Calvert-Lewin's header.
Henrikh Mkhitaryan's third assist set up Ramsey's hat-trick goal with 16 minutes to go as Arsenal moved to within three points of fifth-placed Tottenham Hotspur, while Everton slip to 10th.
Arsenal: 10 Feb v Tottenham Hotspur (A)
Everton: 10 Feb v Crystal Palace (H)