Leeds will also return to the Championship after three seasons in the Premier League as their relegation was confirmed by a 4-1 defeat to Tottenham.
Abdoulaye Doucoure was Everton's hero after the French midfielder smashed home from the edge of the box on 57 minutes to spark jubilant celebrations around Goodison Park.
In contrast, Leicester were crestfallen as they went down, despite their 2-1 win over West Ham.
Everton began the afternoon in control of their own fate but put a raucous support of 40,000 fans at Goodison Park through the mill before securing survival.
Scoring goals has been a problem for Sean Dyche's men and they badly missed the presence of talismanic striker Dominic Calvert-Lewin.
Even when Everton did open up the visitors, Bournemouth goalkeeper Mark Travers produced a stunning save to deny Idrissa Gueye midway through the first half.
Travers was a late replacement in the Bournemouth side after captain Neto missed out for personal reasons and more heroics from the Irish goalkeeper prevented Demarai Gray from opening the scoring early in the second period.
Conceding first would likely have been fatal for Everton, who have not scored more than once at home since October.
The relief around Goodison was palpable when Doucoure's sweet strike finally got the better of Travers.
Everton still had an anxious half hour to see out and Jordan Pickford had to make a fine save to deny Matias Vina an equaliser during 10 minutes of added time to get over the line.
For an optimal experience, please
