Robins said: "We conceded too many poor goals. We knew we had to make changes which we did at half-time. We spoke about not having regrets. We started to look more of a threat.
"To come back from 3-0 down is incredible. Some good quality football. We were really disappointed at the end.
"We hit the bar and had a goal disallowed. It was a really difficult game. We lose Tavares to injury and had to make another change.
"If you can attack them you can really hurt them. I can't be prouder of them. They did fantastically well. I think we've made Coventry proud.
"You get to penalties and it's a lottery. You don't hold anything against anyone if they show the courage to take one in a cauldron."
On the offside goal, he said: "It's a toe nail. Those lines are the bane of our life. You don't get a chance often to go to an FA Cup final and we were really really close.
"We have to pick ourselves up because there's a game on Wednesday.
"There's always a bit of gamesmanship (in penalties). We needed to score the five goals. We scored the five in the summer and missed the sixth and Luton went up. It feels like a kick in the teeth.
"I'm really proud of them.
"We've just said to them they've put themselves right up there in the history of the football club. People will talk about this game for a long time."