The Cherries took the lead early in the second half courtesy of a Joshua King strike but Niasse, who scored a late goal in the EFL Cup win over Sunderland in midweek, was the hero again today with a late brace in the space of five minutes.
The Senegal international was making his first league start for the side since May 2016 and came on for skipper Wayne Rooney, who left the field with blood pouring from his face following a collision with Simon Francis.
The win ended a run of three straight league defeats for the Toffees, and Koeman later admitted to having had "a lot of emotions" in what proved to be an eventful second half.
"[I am] calm after the game, yes," he told Match of the Day. "There were a lot of emotions. It was a difficult game, we did not create many chances and we took the right decision after 1-0 down to try something different - play two strikers and go more direct.
"More credit to the players - Niasse and [Tom] Davies. They did a good job after a difficult time. The boys showed great character.
"[Niasse] is bringing more aggression. All the credit to Oumar Niasse, that is all about life. If you get your chance you need to take it. He took it on Wednesday and again today. He made the difference. He scored two great goals. It was really important that we stood up after going 1-0 down.
"It is a totally different world after winning. You have key moments during the season and this was a big win that will give everyone a boost."
Next up for Everton, who are now 13th in the table, are successive home games against Apollon Limassol in the Europa League and Burnley in the league.