Ramsey hit a hat-trick in a 5-1 win for the Gunners, with Laurent Koscielny scoring as well as debutant and club-record signing Pierre-Emerick Aubameyang.
Fellow new recruit Henrikh Mkhitaryan also shone on his first outing at the Emirates Stadium, laying on three assists as Arsenal ran riot in a first half where the visitors were all over the place.
Toffees boss Sam Allardyce said after the game all of his players were 'crap' after he altered his system to mimic that of Swansea - who shocked Arsenal when they beat them earlier in the week. And Williams sided with Allardyce's opinion as Everton continue on a run of just one win in eight Premier League games.
"Everyone is angry and disappointed with the performance and result," he told the club's official website.
"The manager was not happy - but none of us are. It hurts like any defeat but when you go down like that - 4-0 in the first half - it does really hurt. It is football, and whatever formation you play, or whichever personnel are in the team, you have to do your best.
"You have to compete and we did not compete in the first half. We have used different formations and won, so that is not an excuse."
Allardyce's 500th Premier League game as a manager turned out to be Arsenal's day as the fans were introduced to their two new arrivals.
Mkhitaryan set up Ramsey to open the scoring with just six minutes gone before Koscielny nodded in at the back-post.
Ramsey added a third with a deflected drive before Mkhitaryan fed Aubameyang, who converted coolly despite being offside.
Dominic Calvert-Lewin came off the bench to head a consolation for Everton, before Mkhitaryan teed up Ramsey for his hat-trick strike.
"You can see that he's got electric pace and an eye for goal as well," he said of Aubameyang.
"He was involved in the link-up play as well, so we're definitely happy to have him and we can't wait to see what he brings to the team.
"He's fun. He's learning faces but he's definitely enjoying his time here and there's still more to come from him."
Asked about Mkhitaryan's home debut, Iwobi added: "We've seen glimpses in training and we were just waiting to see it on the pitch.
"As you can see from today, he's just an amazing player. With the quality players we have up front, we're always going to create chances.
"You could see that we scored five today and we created even more chances."
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