Vale are now without a win in four games against their near neighbours. But the visitors were on top in the main and should have made first-half chances for Uche Ikpeazu and Dodds count.
Ikpeazu, who enjoyed two loan spells with Crewe last season, waited too long before getting a shot away after breaking clear and the on-loan striker eventually set up Dodds, who shot straight at goalkeeper Ben Garratt.
Dodds also despatched an excellent cross from Ben Purkiss into Garratt’s arms from close range when it seemed easier to score.
Another former Crewe player, Byron Moore, was a constant thorn in the home side’s flesh with his strong running.
Vale’s on-loan goalkeeper Jak Alnwick kept out a set-piece effort from George Ray in one of the rare moments Crewe posed a threat.
But despite their dominance, Robert Page’s side were kept at bay by some dogged defending, not least by Crewe’s new centre-half recruit Ben Nugent, who stood up well to the physical challenge posed by Ikpeazu.
Ikpeazu did get some space to force Garratt into a low save as Vale looked for a breakthrough in the closing minutes.