Aleksandar Mitrovic scored on his return for Fulham but their defensive woes continued.
The Cherries took the lead through Dominic Solanke with their first attack of the game before Issa Diop levelled from a corner.
Bournemouth re-took the lead before the break when Jefferson Lerma finished past Bernd Leno before Mitrovic’s second-half penalty gave the Whites a point.
It took just over a minute for Bournemouth to take the lead when Solanke played a neat one-two with Philip Billing in the box before hammering a first-time strike past Leno.
Fulham responded well and had their deserved equaliser on 22 minutes when Andreas Pereira’s corner was turned in by the shoulder of Diop to score his first goal for the club.
But Bournemouth re-took the lead just before the half-hour mark when goalscorer Solanke found space down the right-hand side to cut in and tee up Lerma to slot past Leno.
Fulham levelled matters again after the restart when Mitrovic, back in the team after a foot injury, was hauled down by Lerma in the box at a free-kick before sending Neto the wrong way from the spot.
Fulham: Leno, De Cordova-Reid, Diop, Ream, Robinson, Palhinha, Reed (Cairney 45’), James (Willian 45’), Pereira (Vinicius 86′), Kebano (Wilson 62’), Mitrovic