Alex led 3-0 going into this Northern Final second leg and the Sky Blues looked to be heading out with a whimper until a Mark Ellis own goal and Leon Clarke strike - both in injury-time - set up a tense finish.
Davis said: "I'm full of mixed emotions. We gave them a lifeline out of nothing but it's about getting through to our first major final and that's what matters.
"We've taken this competition seriously from the start because we think it's a cup we can win. It's not been easy and this is our reward for taking this cup seriously.
"It's something to look forward to. We had a great time their last year and it will be a great experience for the players, fans and the town.
"I can't praise the players enough for this season. We have consolidated our place in League One and made a cup final, but I'm not going to write off a play-off push.
"Finishing halfway up the league isn't good enough for me and it won't be good enough for the players, we want to finish higher than that."
Coventry's caretaker boss Lee Carsley said: "The fans can go back down the M6 proud of the lads. I've watched a lot of videos lately of great comebacks and they tend to happen late but we left it a little too late this time.
"Leon (Clarke) has said sorry and says he should be more clinical but he doesn't have to apologise to us, he's a good player and when he works as hard as he did for us you can forgive him. He was unplayable.
"The tie was possibly lost at home. It was always going to be an uphill task but it was a task we were up for but fair play to Crewe, they played well and deserve their place in the final."