Burnley won the game 2-0 after a brace by Kevin McDonald.
Coyle told Sky Sports: “We’ve shown that we can play a bit. We’ve shown that there are good sides in the Championship and good individual players.
“A lot was made about the young players Arsenal have, but I had five or six young players out there and that augurs well for the future of our club.
“That’s the standard that we’ve been setting of late and we had to replicate that. We could have very well got to that standard and lost, because of the quality Arsenal have. If you switch off you will be punished.
“We scored two very good goals and could have had a couple more. I’m absolutely delighted, the atmosphere tonight was a real football environment, and I’m delighted that we’ve given everyone a very enjoyable night.
“There’s no doubt that we’ve been big underdogs in all the games against Chelsea, Fulham and Arsenal. The lads have worked their socks off, we knew we had to be resolute and keep a nice shape.
“Brian Jensen has made a few magnificent saves in the game and to reach the semi-final is a magnificent achievement for everyone at the football club.”
He added: “Whoever we get (in the semi-final), we will be underdogs. But we will go and enjoy it. We will only enjoy it by working hard and setting the standards we have previously.
“There’s only ourselves and three other teams that can, so we’ve got a chance.
“There’s no doubt that we’re beyond where we should be, but we’ve done that by working hard and because we have good players and we will look to continue that.”