After two draws and 15 defeats on their travels, Fulham beat a Cherries side that had recorded a 5-0 win at Brighton last weekend, thanks to Aleksandar Mitrovics second-half penalty.
Bournemouth started the match very well and almost broke the deadlock in the fourth minute only for Sergio Rico to make an outstanding save to keep Nathan Akes header out before Ryan Fraser lashed a 25-yard effort narrowly wide of the Cottagers goal.
Fraser then found Josh King who contrived to miss the target from only six yards out when it seemed easier to score.
Rico then tipped the excellent Frasers shot over the bar but, when Fulham responded, Mitrovic set up Ryan Sessegnon whose shot flashed across the face of goal.
Veteran Cherries goalkeeper Artur Boruc got down well to turn Denis Odois header away for a corner as the visitors enjoyed a purple patch just before the interval.
Fulham, who have won two games in a row and kept clean sheets in both of the matches having beaten Everton 2-0 last weekend, continued to be in the ascendancy at the start of the second half and they were awarded a penalty in the 53rd minute when Mitrovic went down under Jack Simpsons challenge. The Serbian got up to beat Boruc with his spot-kick.
After Sessegnon had fired his shot straight at Boruc, Bournemouth almost equalised midway through the second half only for Rico to deny Fraser.
Eddie Howes team pressed for a late equaliser but Rico denied Callum Wilson on two occasions as Scott Parkers team held on for the three points.