Dominic Solanke gave the visitors a flying start for the visitors, combining brilliantly with Danish midfielder Philip Billing and rifling home a first-time finish after 62 seconds.
Mitrovic, recovering from an ankle knock, started slowly but burst into life with a snap shot that was turned behind in the 22nd minute, and Issa Diop brought Fulham level with a thumping header from the resulting corner.
Bournemouth were back in the lead seven minutes later when a defensive lapse from Fulham was punished, Solanke running at Tim Ream before cutting the ball back to Jefferson Lerma who slotted it past Bernd Leno.
Fulham levelled again early in the second half when Mitrovic was wrestled to the ground by Lerma, and the 28-year-old Serbian striker made no mistake from the spot to net his seventh goal of the season.
The sides remained close together in mid-table, with Bournemouth in ninth spot on 13 points and Fulham 10th, one point adrift.