Alexis Sanchez won the game in extra time as the Gunners came from behind to reach a third FA Cup final in four years, having won on their previous two occasions.
Away from the cup, Arsenal find themselves languishing in seventh place and in real danger of missing out on the top four for the first time in Wenger's 21-year tenure.
Ramsey hopes another Wembley triumph will go some way to make amends for poor league form.
"I love this competition," he told BBC Radio 5 live. "We have been quite successful in recent years and you can see the passion that the players play with.
"We went behind and kept on going and got a result. I made the mistake for the goal. I know in these situations you have to respond. I carried on trying to work my socks off and ultimately it paid off.
"That is the second time we played this formation credit, to the boys. We have shown the character we have in this team.
"It was important to win today, to have something to look forward to. But now we have another seven cup finals to try and get into the top four.
"We have let him [Arsene Wenger] down at times this season, we want to win it for him and ourselves."
Danny Welbeck added to BT Sport: "It's been sticky in some patches, obviously in the league we're not where we want to be. We've got to stick it out and hopefully win [the FA Cup]."