Ryan Sessegnon had given the Cottagers the lead in the 70th minute, stabbing home after linking up with substitute Samuel Chukwueze, as Marco Silva's side looked on course to hand the Cherries only their second league defeat of the season.
But Semenyo turned the contest on its head, producing a fine solo equaliser just eight minutes later. The Ghanaian international glided past Timothy Castagne down the left before driving the ball through Bernd Leno's legs from a tight angle.
Justin Kluivert, introduced from the bench, then unleashed a thunderous strike into the top corner in the 84th minute to score his first goal of the campaign.
Semenyo sealed the win deep into stoppage time, finishing a counter-attack started by Ben Gannon-Doak to claim his second of the evening and sixth of the season.
"He is incredible, he shows it every game now," Kluivert told Sky Sports.
"He is just world-class. We are glad to have him – he's the one that scores goals for us and we appreciate that. He's doing an amazing job."
Bournemouth boss Andoni Iraola praised his match-winners, adding: "Antoine and Justin made the difference. The first two goals were amazing goals."
The victory extended Bournemouth's unbeaten league run to six matches, leaving them only one point behind leaders Liverpool ahead of the weekend fixtures.
Fulham, who lined up without a recognised striker due to injuries to Raul Jimenez and Rodrigo Muniz, remain winless on the road and sit 11th with eight points from seven matches.
"It was tough to lose the way we lost it," said Silva. "We scored a great collective goal and the game was completely under control. Until they equalised, they had zero chances. Then two individual moments from two Bournemouth players decided the game. The second one is a great shot from long range, and the first, two against one on the side, we have to be strong in that moment, even if Semenyo is difficult to stop."
Kluivert reflected on the team's mentality, saying: "Losing 1-0 and having the spirit to come back is incredible. It's early in the season, but we have to strive for the highest possible position. The mentality now is not to look down but up, and we will keep doing that."
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