Coventry dominated the first half but could not find the breakthrough and they were made to pay for that by some ruthless finishing by the visitors after the break.
Brad Inman scored twice on the counter-attack to put Crewe in charge before Ajay Leitch-Smith picked Coventry off as they went in search of a goal to cling to ahead of the second leg.
Davis said: "We set up to counter-attack and the plan worked.
"We knew we couldn't match them on the attacking side, we knew we had to be cautious. We watched them. We didn't want it to be open. We can attack but we thought that would be wrong so we didn't plan to do that.
"We felt that was the best thing to do. We have pace with Byron Moore, Brad Inman and Ajay Leitch-Smith who can take a chance so we thought we would play a certain way.
"They are our best finishers at the football club so we didn't want them back defending too much.
"They had to do a lot of defensive work because of the pressure we came under and we led a bit of a charmed life and we rode our luck, which you sometimes have to do in games, but our blocking was superb."