Alexis Sanchez's two goals, the second courtesy of a mistake from Royals goalkeeper Adam Federici, saw the defending champions through after Garath McCleary's leveller for the Championship side.
"We knew it was going to be a tough game, they showed great character to come back into it and we were on the ropes at times,” admitted Ramsey to BBC One.
Reading 1-2 Arsenal
(after extra-time)
The FA Cup Semi-Final
5.20pm, Saturday 18 April
Wembley Stadium Connected by EE
"Obviously we had a bit of luck in the last goal but we're delighted to be going through to another Wembley final.
"Reading made us work for it but we got through in the end. It doesn't matter how, we're in another final."
Federici left the field in tears after fumbling Sanchez's shot through his legs and over the line for the clinching goal, and Ramsey had sympathy for the shotstopper.
"I've felt the experience of losing in a final and it's not a nice feeling. I have sympathy with him but unfortunately a team has to lose and happily for us it was Reading today."
