Alex earned their second win at the Ricoh Arena this month as substitute Max Clayton scored with his first touch.
Leon Clarke had earlier given Coventry a 12th-minute lead that lasted just four minutes thanks to a Mark Ellis volley.
Davis said: "It was a bit scrappy in parts but we came here with a plan again and I thought we wanted to go as long as we could, keeping them out.
"Obviously the early goal was a set back but I always felt we could score from something.
"The game was fairly even. It was a bit of a battle and scrap and that's maybe the result of playing each other four times.
"I think that makes a difference. You know each other player-wise and although we tried to change the shape it is still difficult because our paths have crossed three times already so it was difficult."
Crewe keeper Steve Phillips was man-of-the-match on his last visit to the Ricoh Arena and enjoyed himself again.
"He made two or three important saves, as we expect him to do because that's what he is there for," added Davis.
While the message to Clayton was clear. Davis said: "I told him to go out there and score the winning goal. It's nice when you have a talented 18-year-old to bring on and get you a goal."
