After a cautious start from both sides in the opening stages, the match suddenly kicked into life after 11 minutes when a goal from Dominic Calvert-Lewin was retrospectively ruled out for a handball by VAR to the huge relief of the Cherries supporters.
Despite that setback for Leeds, they kept the pressure on the hosts’ defence, with a shot from Harry Wilson in a promising position narrowly missing the target on the quarter-hour mark.
However, Bournemouth took the lead in the 31st minute after capitalising on a dreadful error from James Trafford.
A poor pass from the England international goalkeeper inside his own box allowed Justin Kluivert to play a square pass to Evanilson, who made no mistake, slotting it into the back of the net to secure his fourth league goal of the season.
That mistake seemed to really knock the stuffing out of the West Yorkshire outfit, and they created little in the way of clear-cut chances to try and level the match before the interval.
Both sides knew how crucial the next goal would be, so the second half began cagey, with very little goalmouth action.
The first noteworthy opportunity after half-time came in the 63rd minute as Trafford was forced to palm away Marcus Tavernier’s free-kick from a decent position.
A curiously pedestrian second half continued to produce a very flat spectacle, with James Justin’s header into the side netting from around the far post marking one of the few highlights.
With Leeds throwing everyone up forward in search of an equaliser in the fifth minute of stoppage time, Marco Rose’s side killed the game off with a goal at the death from substitute Amine Adli from eight yards out.
After this result, Bournemouth have recorded six successive clean sheets in all competitions, and their latest win has moved the Cherries into the top half of the Premier League.
They must now shift their attention to the league phase of the UEFA Europa League when they travel to AC Sparta Prague on Thursday.
Meanwhile, Leeds are without a win in their last five matches, which has seen them drop to ninth, with a league game against Tottenham Hotspur at Elland Road next on their agenda.
Flashscore Man of the Match: Ryan Christie (AFC Bournemouth)