Mark Ellis grabbed the decisive goal six minutes from time as the Railwaymen, promoted to npower League One last season via the play-offs, made it three wins in four games.
"I wanted us to be on 30 points at this stage, but we have exceeded that target by six points," said Davis. "We are in a good league position and we will keep pushing ourselves to make sure we keep improving.
"We'll lose games and have our bad days, but it's important not to get disheartened. You have to build consistency and put little runs together, which is what we are trying to do.
"We lost at Tranmere on Boxing Day but there was no need to panic because there wasn't much between the teams. Today epitomised this league because we had to battle for anything we got. It's encouraging that we bounced back with a win.
"The game was scrappy but both sides tried to play football in difficult conditions. We came with a plan to deal with Oldham's big lad up front (Matt Smith) because he was bound to be a threat. It was a bit of an onslaught at times, but we did that."
Crewe went in front after 61 minutes as Mathias Pogba's 20-yard blast took a deflection and flew inside the right-hand post.
Oldham levelled after 82 minutes when substitute Dan Taylor seized on a defensive mistake and drove a shot into the far corner from 12 yards out but Alex regained the lead within two minutes, Ellis converting a close-range header from Byron Moore's cross.
Davis added: "I'm pleased for Mathias Pogba because he missed a great chance early on and, at half-time, we told him not to worry about it. He was a bit fortunate to get a deflection on his goal, but you don't score if you don't have a go.
"Mark Ellis made a mistake for Oldham's goal when he slashed at the ball, but all credit to him for getting in there for the winner."