Ramsey completed his treble in the second half from Mkhitaryan's cutback, and Dominic Calvert-Lewin headed in a consolation.
Moments later Cech was down and receiving treatment which forced the Gunners to make a substitute.
The experienced stopper picked up a knock in the 69th minute on Saturday night and had to be replaced by David Ospina.
Arsenal manager Arsene Wenger has confirmed that the club will need to run tests on the calf injury sustained by Petr Cech in order to determine the severity of the problem. I don't know medically how bad it is. Petr Cech goes down after sustaining an injury.
- Pierre-Emerick Aubameyang became the eighth player to score on his first ever Premier League appearance for Arsenal, after Steve Bould, Kevin Campbell, Freddie Ljungberg, Emmanuel Adebayor, Samir Nasri, Thomas Vermaelen and Alexandre Lacazette.
Arsenal fans were chanting "Wenger In" tonight after a small protest outside the stadium fell flat on its face.
Laurent Koscielny got in front of Aubameyang to head the second at the far post after Shkodran Mustafi glanced on Mesut Ozil's corner before Ramsey scored a deflected third.
A delight for Wenger but Everton boss Sam Allardyce was left fuming.