The Alex had only picked up one point from their opening two league games, but two strikes from teenager Ryan Colclough either side of half-time earned the Gresty Road side three points.
Winger Colclough opened the scoring after 41 minutes when he skipped his way past Tranmere defender Ash Taylor before smashing into the top corner from inside the box.
The 18-year old then doubled his and Crewe's total for the afternoon when he tapped home Byron Moore's shot from close range with seven minutes left.
Manager Davies was delighted withthe youngster's contribution.
"I thought he was good," he said. "We keep saying about him not scoring and needing to be a bit more selfish.
"He caused them problems and got them on the back foot.
"They couldn't cope with his quick play or quick feet. He was a menace."
Colclough's goals were not only his first for the season, but also his first at Gresty Road.
But it was Crewe's other winger, Moore, who replaced Ajay Leitch-Smith in the starting line-up, who troubled the visitors the most in the first half, as he played a key role in the dismissal of Tranmere's Danny Holmes.
The Rovers right-back was dismissed for two yellow cards, the first of which was received after just 11 minutes when he pulled down Moore.
And then little more than 20 minutes later, his tackle from behind on the same man earned him his marching orders.
The Alex made their one-man advantage pay and could have enjoyed an even greater winning margin.
Substitute Max Clayton hit the post with a drive from inside the area, while Colclough was denied a hat-trick when his fierce shot from a tight angle stung the fingers of goalkeeper Owain Fon Williams.
Davis said Tranmere could have no complaints about the sending off and was pleased with how his team coped with the man advantage.
"If we'd only got the draw, it would have been a travesty, because they were down to 10 men," he added.
"But it was good to get the win and it's always pleasing to get the first three points.
"It gives us something to build on for the next game. Byron Moore got brought down two or three times.
"And the second one, I don't think he had too many complaints. It was the right call to send him off.
"I thought we went about the game well after the sending off, but in the second half we didn't put the game to bed.
"We were a little bit slow in the passing. But I'm always pleased to win. We can now enjoy the week."