All of the goals came in a topsy-turvy first half which saw Jermain Defoe break the deadlock after only 10 minutes and a late flurry of goals just before the interval.
Luka Milivojevic and Scott Dann scored in quick succession to turn the game around in the final five minutes of the half, only for Defoe to double his personal tally and restore parity deep into first-half stoppage time.
There was more stoppage-time drama in the second half too when Palace were awarded a penalty which could have put them level on points with 17th-placed West Bromwich Albion, but Christian Benteke missed from the spot as Roy Hodgson's side were forced to settle for a point which instead leaves them bottom of the table.
Palace immediately set about looking for an early opener and made a bright start to the match, with Andros Townsend, Jeffrey Schlupp, Wilfried Zaha and Ruben Loftus-Cheek all posing an attacking threat inside the opening 10 minutes.
It was Bournemouth who got the early goal against the run of play, though, as Junior Stanislas and Andrew Surman worked a clever corner routine before finding Defoe, who fired an unerring finish into the bottom corner.
Palace continued to push forward, though, and Asmir Begovic was needed to deny Benteke in the 16th minute before Bournemouth survived a mini-goalmouth scramble moments later.
The Eagles were again left rueing a lack of presence in the six-yard box shortly after the half-hour mark when Benteke nodded Zaha's free kick down in a dangerous area which no Palace player could capitalise on to add the finishing touch.
The home side's pressure finally paid off with just four minutes remaining of the first half when Zaha was brought down by Begovic inside the penalty area, and Milivojevic stepped up to fire the resulting penalty into the top corner, restoring parity in the process.
Palace were in no mood to settle for a draw, though, and just three minutes later they had turned the game on its head when Dann - on for the injured Mamadou Sakho earlier in the game - slid in at the back post to nudge the ball over the line for his fourth goal in his last four games against the Cherries.
There was still time for more drama in an action-packed end to the first half, though, and Defoe produced the pick of the goals with virtually the last action of the half when he lashed a stunning half-volley past Julian Speroni from an acute angle.
Defoe had a chance to complete his hat-trick shortly before the hour mark when he dragged his shot wide of the target from inside the box, and Bournemouth came close again 10 minutes later when Jordon Ibe drew a smart save from Speroni.
Palace should have regained the lead with just under 15 minutes remaining when Dann was gifted a glorious chance to further improve his remarkable goalscoring record against Bournemouth, only to fire over from close range having been teed up by Benteke.
It was Palace who continued to knock on the door late on, though, with Loftus-Cheek and Milivojevic both coming agonisingly close before the hosts were gifted the best chance of the lot in stoppage time.
Despite Milivojevic having already scored once from the spot, Benteke took on the duties second time around and it proved to be a mistake as Begovic atoned for giving away his own penalty earlier in game to thwart the Belgian from 12 yards.
Begovic was needed to make one more routine stop from Milivojevic in the dying stages of the game, but Bournemouth held out for a point which keeps them 14th in the table, although they are now winless in their last four.
Palace, meanwhile, are unbeaten in five games now but sink to the bottom of the table courtesy of wins for both West Ham United and Swansea City.
Crystal Palace (4-4-2): Speroni; Fosu-Mensah, Tomkins, Sakho (Dann, 25'), Schlupp; Townsend, Cabaye (McArthur, 65'), Milivojevic, Loftus-Cheek; Zaha, Benteke
Bournemouth (4-4-1-1): Begovic; Francis, S Cook, Ake, Daniels; Stanislas (Ibe, 65'); L Cook, Surman, Fraser (Afobe, 87'); King; Defoe (Wilson, 80')