Everton, who were winless at Goodison Park this season, pounced on an early error from defender Illia Zabarnyi to take the lead in the eighth minute.
Later, a decision from goalkeeper Neto to punch the ball and clatter into his own teammate also cost Bournemouth as Jack Harrison scored from distance. Everton did not look back to take three points and leave Bournemouth 19th in the table.
"Until we are more solid defensively we will make those mistakes and it will be tough to win games. We knew that whoever scored first would be a big advantage," Andoni Iraola told reporters.
"Mistakes are pretty obvious, it's not the first time that's happened this season and it's costing us points, games. We have to accept there is a moment when you have to take responsibility. We are a team but they are costing us points.
"We knew it would be a huge game, very important for us, it changed the situation a lot from winning or losing this game. We made the mistakes and they didn't."
Bournemouth struggled to make inroads into Everton's box and the atmosphere at Goodison Park caused the visitors to become nervous as Sean Dyche's side piled on the pressure to win the ball back several times.
"Concentration is something we have to work on in training. You have to focus for 100 minutes and not make these mistakes," Iraola added.
"The same as today, when you're in relegation spots you want to take the points. We had chances today but we didn't take it. We needed to be concentrated and we were not. We needed to be solid at the back and we weren't.
"We have to find the way and turn this around."
