The Toffees made three changes, with Ademola Lookman making just his second league start of the season, while Nathaniel Clyne made his league debut for the Cherries, who were without star striker Callum Wilson.
It was the visitors who made the better start, David Brooks striking the foot of Jordan Pickford's right-hand upright after nice interplay between Josh King and Junior Stanislas and the latter also forced a smart save with a dipping free-kick.
Everton were poor in the opening 35 minutes or so but a decision to switch Lookman to the left sparked their attack and the young winger was thwarted by a great block by Clyne at the back post.
Michael Keane then skidded a header off the crossbar, while Sigurdsson fired wide as the Toffees took charge heading into the break.
Bournemouth had the first chance of the second half when Steve Cook glanced a header onto the roof of the net but Everton opened the scoring just after the hour.
Lucas Digne was the architect, digging out a cross from the left and Kurt Zouma nodded home unmarked at the near post.
The Cherries tried to respond as Josh King fired wide but the visitors regularly fell short in attack, despite Eddie Howe introducing Lys Mousset and Jordan Ibe.
Richarlison had a clever header wide of the upright for the hosts but it was substitute Dominic Calvert-Lewin who doubled the advantage in stoppage time.
The excellent Lookman was again involved, finding his team-mate who coolly fired home from the edge of the box.
That sealed Everton's first home league win since November, moving them up to 10th in the table while Bournemouth's tough afternoon sees them drop to 12th.