Adam Murray’s players conceded twice in as many first-half minutes before Nick Haughton and Gold Omotayo saw Fylde level with 12 minutes to go.
Chris Conn-Clarke’s penalty on 87 minutes looked to have given the visitors victory before Davis was the Coasters’ late hero.
Murray made four changes from Saturday’s game at Hartlepool United with Davis, Taelor O’Kane, Danny Whitehead and Siya Ligendza replacing Alex Whitmore, Connor Barrett, Jon Ustabasi and Omotayo.
Josh Kay had Fylde’s first sight of goal, seeing his low effort kept out by Ethan Ross, who also held Kieran Glynn’s attempt.
At the other end, Davis blocked Regan Linney’s effort and Dior Angus volleyed over before Altrincham led with 12 minutes of the first half remaining.
Theo Richardson parried Angus’ fierce shot and Linney reacted quickest to follow up and score.
They doubled their advantage two minutes later when Angus, unmarked on the edge of the box, drove the ball into the bottom corner.
Haughton dragged a shot wide before half-time, after which the Coasters searched for a response.
They were awarded a penalty on 57 minutes after Jake Cooper fouled Haughton, who saw his spot-kick saved.
However, from the rebound, Whitehead was fouled and Fylde earned another penalty which Haughton converted.
Fylde grew in confidence with Ustabasi and Omotayo sent on in quick succession, replacing Owen Evans and Glynn.
Ustabasi almost made an immediate impact, having a header kept out by Ross, before Whitehead’s well-hit strike was off target.
Ross also punched away a Haughton free-kick before Fylde levelled on 77 minutes as Omotayo headed in from close range.
Ten minutes later, Emeka Obi brought down Justin Amaluzor and Conn-Clarke beat Richardson from the spot.
An unbelievable game saw a spectacular finale as Davis gave Ross no chance with a thumping header to level matters.
AFC Fylde: Richardson, Conlan, Davis, Obi, Evans (Ustabasi 63), O’Kane (Philliskirk 80), Glynn (Omotayo 66), Kay, Whitehead, Haughton, Ligendza. Subs not used: Whitmore, Barrett.
Altrincham: Ross, Banks, Eddy Jones, Baines, James Jones, Osbourne, Linney (Marriott 58), Angus (Amaluzor 83), Cooper, Wilson, Joey Jones (Conn-Clarke 79). Subs not used: Roscoe, Burroughs.