It took Aneke just five minutes to open his account at Priestfield, although Amine Linganzi levelled for the hosts on the half-hour mark.
Aneke put Crewe back in front with his 16th goal of the season and Mathias Pogba wrapped up all three points for the visitors late on.
Crewe ended the game with 10 men as Anthony Grant picked up two yellows in quick succession in the dying minutes, but it did little to dampen the mood as a first win in six games lifted them out of the League One relegation zone.
Manager Davis was full of praise for Aneke, insisting his double is just the tip of the iceberg for the 20-year-old.
"I though Uche [Ikpeazu] and Chuks were tremendous in the way they battled against the centre-halves," Davis said.
"Chuks took his goals tremendously well. You can see the qualities he brings to the team. He gets in great positions, he's got two great feet, he's got great strength and great balance.
"He works very hard on his game, he does extra work each day, and you can see the rewards he's getting.
"It wasn't pretty at times, but we defended when we had to. Our play was good, they gave us more time on the ball than we expected and we made good use of the ball.
"We're in control of our own destiny, we've got enough games to do it, we've shown what we're about here and hopefully we can repeat that next time out. It's a hard league to stay in, but we're determined to."
- Sports & Recreation
- Mathias Pogba