And Adams praised the character of his side for coming from behind to grab the win.
Jennison Myrie-Williams' last-gasp strike handed the visitors all three points as he snuck in at the back post to tap home a cross that ran right along the Alex six-yard box.
The defeat extends Crewe's winless run to seven matches and piled more pressure on boss Davis as his side sunk deeper into the Sky Bet League One relegation zone.
"I've got a good dressing room," Adams said.
"The lads are working as hard as they possibly can to establish the side in the First Division, hopefully we'll be on onwards and upwards.
"But we can never take it for granted.
"We have to perform, come out, roll our sleeves up and scrap every game.
"We're not Real Madrid and we don't make any apologies for the way we play - we get balls into the box and if that puts people under pressure and they don't like it, we'll keep doing it.
"I knew we'd get stronger, we pride ourselves on our fitness and work ethic and we kept going and gradually wore Crewe down."
A tense start saw Crewe have the best of the chances as they pushed for the first goal in front of a bumper Gresty Road crowd.
And they took a deserved lead through a stunning strike after 34 minutes from Manchester City loanee George Evans.
It was a goal of the season contender from the teenager, who received the ball 25 yards from goal and only needed one touch to send home a stunning side-footed volley that dipped perfectly over Vale keeper Chris Neal and into the top corner.
Alex had earlier come close to breaking the deadlock after 19 minutes, with Evans again involved as he curled against the post from 22 yards.
As full-time loomed Myrie-Williams popped up at the back post to easily net and send an army of over 2,000 Vale fans into jubilation.
- Sports & Recreation