Adam Leitch-Smith gave Alex the ideal start when he fired home a brilliant 35-yard effort in the first minute before Luke Murphy snatched a winning goal after Mike Grella had levelled the scores.
Davis' men suffered a home defeat to Notts County on the opening day, but their manager was pleased with his team's response.
He said: "For us, it was important to get up and running with our first points on the board.
"It's going to be a tough season for us, we know that.
"I thought we played better on Saturday when we lost at home to Notts County than we did here. But maybe that is what this season is going to be like."