Fulham, who will make a return to the Championship next season, had lost 15 and drawn the other two of their 17 away games this season.
The goal came in the 53rd minute when Bournemouth's Jack Simpson fouled Mitrovic inside the box and the Serbian striker stepped up to slot home the resulting penalty.
The win was the Cottagers second straight following last week's 2-0 win over Everton.
.@FulhamFC recorded their first away win of the season as an Aleksandar Mitrovic penalty secured a 1-0 victory at Bournemouth on Saturday.— Sky Sports Premier League (@SkySportsPL) April 20, 2019