The width of the crossbar denied Nathan Ellington the glory of making the swiftest of impacts on his debut as the Alex pressed hard in the closing stages of an intriguing derby battle.
Shrewsbury were a cut above Davis' side when they secured automatic promotion from League Two last season but the Railwaymen boss believes his men have adapted better to League One.
The current seven-point gap between the two sides backs that up, yet Luke Summerfield's first-half equaliser secured a point for Shrewsbury to add to their earlier 1-0 victory at the Greenhous Meadow.
"We were better than them in and we have been in all three games and that's no disrespect to them," said Davis.
"They were better than us last season even with Nick Powell and Ashley Westwood in our team. But we're a much better all-round team now and that performance was good enough to win the game."
In truth, there was little to separate either side in a good quality tussle and a much better one than expected on such a scarred and barren surface.
Both teams kept the ball down in the opening exchanges with Alex skipper Luke Murphy blasting wide after a neat interchange.
But the Railwaymen were flying into a tenth-minute lead after some simple attacking play caught out a square Shrewsbury defence.
Ajay Leitch-Smith slipped a ball inside for Byron Moore, who finished with aplomb, right-footed into the far corner in the 10th minute.