Wenger had delayed putting pen to paper on a new contract, despite stating publicly that he was planning to stay.
The Frenchman’s contract at the club expires in the close-season, but he has repeatedly assured both the club and their supporters that he would sign a new deal.
Ramsey scored winner to give Arsenal their 11th FA Cup, equalling the all-time record held by Man Utd.
The title race is realistically down to Manchester City and Liverpool now, after Chelsea’s latest stumble.
Arsenal will be assured a fourth-place finish if Everton lose at home to Manchester City on Saturday.
With two Premier League games remaining, Arsenal have 73 points, four clear of fifth-placed Everton.
Wenger fielded a team with average age of 29, with Olivier Giroud their youngest outfield player at 27.
Arsenal lost their hold on fourth place when Everton defeated Sunderland 1-0 on Saturday.
After extra time ended 1-1, Arsenal’s players scored all of their penalties and Lukasz Fabianski made two vital saves in a 4-2 shootout victory over Wigan.
Arsenal beat Wigan Athletic 4-2 on penalties and reach FA Cup final for first time in nine years.
With five matches to play Everton have 66 points, two more than Arsenal.
Arsenal now face a buoyant Championship side in Wigan, who stunned Manchester City in last year’s final.
Arsenal, without a trophy since their 2005 FA Cup win, have suffered alarming slump in form in recent weeks.
Arsenal’s Cup form is in contrast to their dismal league form which has seen them slide down to fourth spot.
Arsenal have a number of injury concerns as they prepare for Sunday’s north London derby at Tottenham Hotspur.