Josh King's goal could have been the platform for a win had Jermain Defoe not been denied by Jordan Pickford shortly after.
Rooney challenged for a cross in the Bournemouth penalty area, but ended up getting a smack of Simon Francis' elbow.
And Koeman, whose side were below-par for much of Saturday's home encounter, was quick to acknowledge the importance of Niasse's display.More news: USA considers shooting down NK missiles that don't pose a direct threat
He was forced into a formation change before the game, with centre-back Michael Keane on crutches with a gashed foot and Phil Jagielka nursing a hamstring injury which will keep him out until after the worldwide break, prompting a switch to four at the back.
He struck twice in the last 13 minutes to lift his team out of the bottom three and the pressure off under fire Ronald Koeman. The Cherries now languish at 19th in the table, one point from safety. Easily evading Holgate's despairing slide tackle, the Bournemouth striker advanced unimpeded towards goal and drilled a low shot into the corner to make it 1-0.
It proved an inspired change as Davies set up Everton's 77th minute equaliser with a precise cross that his fellow substitute Niasse slammed into the net.
'If we had gone 2-0 down it would have been really hard'.More news: Kylie Jenner and Travis Scott expecting
Parity was restored, however, 1 1 minutes from time when Niasse struck his first.
"The doctor said to me at half-time if it happened again we would need to change him; I changed him because we needed something different". He came on and it was better for Dominic (Calvert-Lewin) to have somebody around him. "Work hard and get your chance.' That is what he did, his attitude was ideal".
It's hard to see the Blues kicking on back towards challenging the top six if he persists with chopping and changing while returning to the same formation and personnel for League games. Some Blues have been calling for a more conventional 4-4-2 formation, particularly at home, but whether Koeman is ready to eat the entire humble pie just yet remains to be seen. Everton fans have been crying out for a new goalscoring hero since Lukaku's summer move to Manchester United.
Niasse wasn't content to settle for his first Premier League goal for Everton and he bagged the victor in the 82nd minute.More news: Japan Open: Solid starts by PV Sindhu, Saina Nehwal
The team of course looked like they are struggling for confidence.