Arsenal lie sixth in the Premier League, eight points behind Chelsea in fourth after suffering a 3-1 loss at struggling Swansea City in midweek.
Shortly after the hour mark substitute Dominic Calvert-Lewin reduced the deficit with a header, his eighth goal of the season, but Ramsey completed his hat trick on 73 minutes to round off the win.
Shkodran Mustafi met Mesut Ozil's corner with a header towards the far post, where Koscielny stooped to head home from close-range.
Yannick Bolasie and Morgan Schneiderlin both started, with Wayne Rooney, Tom Davies, Phil Jagielka and Gylfi Sigurdsson all dropping to the bench.
Mkhitaryan's pass split the Everton defence and, although Aubameyang looked offside, the linesman's flag stayed down as the debutant clipped a deft finish over Pickford.
The Gunners have beaten Everton 96 times in league competition, more than one side has ever beaten another in Football League history.
Mkhitaryan was inches away from marking his home debut with a goal in the 9th minute, after his shot from 20 yards out on the left flew just wide of Pickford's goal.
Pierre-Emerick Aubameyang became the eighth player to score on his first Premier League appearance for Arsenal, after Steve Bould, Kevin Campbell, Freddie Ljungberg, Emmanuel Adebayor, Samir Nasri, Thomas Vermaelen and Alexandre Lacazette.
No Arsenal player has supplied more assists in a single game since Santi Cazorla created four goals against Wigan Athletic in May 2013.
Everton's dismal display was ruining Sam Allardyce's 500th match as a Premier League manager and there was worse to come in the 19th minute.
"His assists were there and we know he's got that quality".