The competition is Arsenal's last hope of silverware in what still could be Wenger's final season at the club, having suffered Champions League elimination in the last 16 and languishing seventh in the Premier League.
With Wenger yet to publicly commit to a reported two-year deal that is thought to have been offered to him, Ramsey hopes helping Wenger to a record seventh FA Cup triumph can go some way to making up for a vastly underwhelming campaign.
"We've let him down at times this season," Ramsey said.
"But he's kept believing in us and the quality we have and we wanted to get to the final and now hopefully we can go on and win it for him and for ourselves.
"I absolutely love this competition. We've been quite successful in recent years and you can see the passion that the players play with."
Sergio Aguero's second-half goal initially looked to set City on their way to a ninth successive Cup semi-final win, but Nacho Monreal equalised before Alexis Sanchez struck decisively in the 101st minute.
Having conceded possession to Yaya Toure, who assisted Aguero, Ramsey was relieved.
"I made the mistake for their goal, that was painful for myself," said Ramsey, who scored Arsenal's extra-time winner against Hull City in the 2014 final.
"But I know in these situations that you have to respond to it and you can't let these things get to you.
"I carried on going and trying to work my socks off, get into position and tackle and ultimately it paid off."