The Cherries were heading to their first league defeat since losing to Liverpool on the opening night of the season when Ryan Sessegnon put the visitors in front on 70 minutes.
However, Semenyo’s brilliant individual run and finish kickstarted the comeback.
Justin Kluivert blasted into the top corner before Semenyo completed the turnaround on the counterattack deep into stoppage time.
Andoni Iraola’s men closed to within one point of leaders Liverpool, who travel to Chelsea on Sunday.
Bournemouth’s start is even more impressive given a number of their better performers last season were prized away for big fees during the summer transfer window.
Defenders Dean Huijsen, Milos Kerkez and Illia Zabarnyi departed for Real Madrid, Liverpool and Paris St-Germain respectively.
“It’s a great feeling,” said Semenyo on sitting second in the table.
“We lost so many important players for us, but the start to the season has been great and it’s a great environment to be a part of.”
Sparked into life
At the Vitality Stadium, neither side was capable of playing flowing football in awful weather conditions as Storm Amy battered the UK with heavy rain and ferocious winds.
But the game came to life in the final 20 minutes after Sessegnon exchanged a slick one-two with Samuel Chukwueze before firing high past Djordje Petrovic.
The lead lasted only eight minutes as the in-form Semenyo burst past Timothy Castagne to reach the byline and then produced an inch-perfect finish through the legs of goalkeeper Bernd Leno.
Semenyo was also credited with the assist for the second, but Kluivert did all the hard work with a stunning strike from distance.
The Dutch international has largely been used as a substitute by Iraola at the start of the campaign despite scoring 13 goals last season.
But Kluivert showed his quality as he collected possession from Semenyo just inside the Fulham half and strode forward before unleashing a dipping drive into the top corner.
In their quest for an equaliser, Fulham were caught short at the back in stoppage time.
Ben Gannon-Doak broke clear from a Fulham corner and kept his composure to wait for support before supplying Semenyo to score his sixth goal in seven Premier League games this season.
Defeat leaves Fulham still in 11th with eight points from their opening seven games.
“It’s very tough for us the way we lost,” said Fulham boss Marco Silva.
“The game was totally under control at 1-0.”