The Gunners moved back above bitter north London rivals Tottenham into fourth place with an Emirates Stadium victory which condemned the FA Cup winners to relegation on Tuesday night.
Lukas Podolski scored twice with Theo Walcott and Ramsey also on target as three goals in eight devastating second-half minutes settled the contest and made Shaun Maloney's first-half effort nothing more than a consolation.
Wales international midfielder Ramsey now has his sights set on Sunday's clash at Newcastle, where victory will ensure another top-four finish for Arsene Wenger's men.
Ramsey told Sky Sports: "We got the early goal and then we sort of panicked a bit. They showed what good form they have been in of late when they got a goal back.
"In the end, we did the job, we got the win and it's the perfect way to finish the season here. We have one game left to finish off to get that Champions League place.
"This is what we love to do - play football. Every game there is a lot of pressure, this game was no different tonight, we had to win to keep ourselves in the driving seat and we managed to do that.
"We are looking forward to going to Newcastle, finishing the job off and getting into the Champions League next season."