Granit Xhaka's casual pass to Ramsey was masterfully flicked past Schmiechel - two minutes later, the attacker Olivier Giroud, who came in the 67th minute, scored the winning goal, lifting the Gunners to glory.
It was striker's first shot for his new club and he now becomes just seventh Arsenal player to score on his debut.
Everton are known to in the market for a new striker this summer, and following Arsenal's £52m purchase of Alexandre Lacazette, Giroud was heavily linked with a move to Goodison Park where he would be more likely to be the main striker.
The subs paid off, with Ramsey scoring in the 83rd minute to make it 3-3 and Giroud heading in off the post two minutes later to win a thrilling encounter for Arsenal.
"But I don't want him to go and in the end he chose to stay and it was one of the days of recent weeks for me because I know how important he is".
'We wanted to do that today - unfortunately that didn't happen and it's something where we can look back and analyse the game and see where we went wrong and improve on the situation'.
Giroud appears willing - if a little reluctantly - to accept his role as an understudy to Lacazette. He'll play 30 games, he'll give you 15 goals.
Wenger further reiterated the point in his written press conference, describing Giroud's decision to remain with Arsenal as a "relief" and "one of the greatest days I've had in recent months".
Tonight's victor was his 70th Premier League goal - coming at the start of his sixth season. I feel as well that the goals we conceded, we can work together to get rid of that because one was a corner, one was a ball we lost in the build up and the third goal was on a corner again, so we conceded two goals on corners.
"Insecurity goes quickly through a team and we had many experienced defenders out tonight - Koscielny, Mustafi, Mertesacker". I don't know whether it was a lack of concentration or game management.