Hull was struggling when manager Marco Silva produced a masterstroke by sending on Niasse just after the hour mark - and the striker responded with two goals inside 10 minutes.
Oumar Niasse stepped off the substitutes' bench to score twice for Hull and while Alfie Mawson ensured a nervous finish it was not enough for the Swans, who saw Martin Olsson and Fernando Llorente injured. We work two systems and the players know what we want.
The win puts Everton in 7th place with 47 points, equalling their total points from the entire of last season, with 10 games left to play.
Marauding right back Omar Elabdellaoui broke clear and passed to Abel Hernandez, who flicked the ball on first time for Niasse to run on and steer a low shot beyond goalkeeper Lukasz Fabianski.
"Two injuries in the first half have hurt us and in the second half we were limited to what we could do", he said.
Silva was aware going into the game that the pressure was on his side to earn three points with home advantage against a relegation rival.
Keen to play a fluid passing game, the Portugese manager admitted:"I've said to our board that it's impossible to play good football".
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"I think from our point of view they were two sloppy goals to give away".
"It was our obligation to win this game and it was a very tough one for us".
Bournemouth will be without Tyrone Mings when they host West Ham United on Saturday as the defender begins a five-match ban after being charged with violent conduct for a clash with Manchester United forward Zlatan Ibrahimovic last weekend.
Andre Ayew thought he had grabbed a point for the Hammers seven minutes from time but King pounced late to complete his hat-trick and nudge the Cherries closer to safety.
King then made up for his penalty miss by bringing Bournemouth level again in the 31st minute before Benik Afobe had Bournemouth's second penalty saved by Darren Randolph. At home we need three points from every game.
Martinez's replacement Ronald Koeman liked Niasse even less and he was sent to play with Everton's under-23s.
"It sums up the Premier League really - the ups and downs are extreme but I think we played well".