It was a flat performance from Bournemouth, who have now lost four of their past five games and have not won in that time.
The first half saw few chances for either side, with the Cherries seeing a goal disallowed on the stroke of halftime as Junior Kroupi was offside when he finished.
It was a similar second period, but Everton looked a threat to the Bournemouth back line and broke the deadlock when Grealish found the bottom corner from outside the box via a deflection off Bafode Diakite.
The hosts at Dean Court weren’t able to find an equaliser in the remainder of the game and were beaten.
Andoni Iraola named the youngest Cherries Premier League XI at 23 years and 254 days as he made four changes from the side which was beaten at Sunderland.
Alex Jimenez _(Image: Richard Crease)_
Veljko Milosavljevic came in for the suspended Marcos Senesi, while Alex Jimenez, Justin Kluivert and Junior Kroupi replaced Adam Smith, Marcus Tavernier and Evanilson, who all dropped to the bench.
Everton saw more of the Cherries goal in the opening exchanges, without any clear chances.
The best opening fell to centre back Jake O’Brien after a Vitaly Mykolenko long throw dropped to him in the box, but his effort was comfortably blocked by Alex Scott after 13 minutes.
An Everton corner was headed out as far as James Garner by Scott, and the ball was chested on to Tim Iroegbunam, who shot well over the bar.
Justin Kluivert _(Image: Richard Crease)_
The best Cherries chance in the opening 25 minutes fell to Antoine Semenyo.
He powered goalwards after O’Brien failed to clear an Adli free kick from the left and it dropped to him, but Jordan Pickford made a good reaction save.
There was a brief look to see if O’Brien handballed it as the ball brushed the top of his arm, but nothing was given.
Everton threatened with their height towards the end of the first half from three successive corners.
Adrien Truffert and Scott glanced headers over their own bar from the first two, the latter clipping the top of the woodwork, before Thierno Barry threatened at the front post but Djordje Petrovic got there first to punch clear.
James Garner shot comfortably over the bar and the Steve Fletcher north stand, before the Cherries had a goal disallowed with the final act of the first half.
Junior Kroupi had a goal disallowed. _(Image: Richard Crease)_
Jimenez crossed from the right and Adli flicked it on to Kroupi, who finished well past Pickford at the near post, but the Frenchman was offside.
The start of the second half continued as the first left off, with few openings for either side.
Tyler Adams was booked for his fifth of the season, before the Toffees went close.
Kiernan Dewsbury-Hall drove into the Cherries half and the defence wasn’t able to deal with his ball, it deflected to Carlos Alcaraz who turned it onto Barry, who went in on goal but Petrovic was able to come out and save past the post.
Junior Kroupi _(Image: Richard Crease)_
Iraola brough Evanilson and Tavernier on for Kroupi and Adli on the hour mark, and the Brazilian was immediately involved.
After Alcaraz had a shot blocked, the Cherries raced forward through Kluivert. The Dutchman squared it to Evanilson who flicked it on neatly to Semenyo, but his shot was straight at Pickford.
Milosavljevic headed over the bar from a Kluivert cross following a corner, before Barry smashed well over the top at the other end moments later.
Julio Soler replaced Kluivert with 20 minutes to play, and Everton opened the scoring eight minutes later.
Djordje Petrovic after the Everton goal. _(Image: Richard Crease)_
The Toffees broke at pace and Alcaraz played it into Grealish, and the Englishman’s shot from 20 yards deflected off Diakite and into the bottom corner.
Enes Unal replaced Scott after the goal, before Iliman Ndiaye drew a save out of Petrovic shortly afterwards.
The Cherries tried to throw everything at their visitors, but David Moyes’ side held firm and kept them at bay to secure all three points.
**AFC Bournemouth**: 1, Đ. Petrović; 3, A. Truffert; 18, B. Diakité; 44, V. Milosavljević; 20, Álex Jiménez; 8, A. Scott (Unal 80’); 12, T. Adams (booked, 53’); 24, A. Semenyo (c); 19, J. Kluivert (booked, 45’)(Soler 71’); 21, A. Adli (Tavernier 60’); 22, J. Kroupi (Evanilson 60’(booked, 89’)).
**Everton:** 1, J. Pickford; 16, V. Mykolenko (booked, 50’); 6, J. Tarkowski (c); 15, J. O'Brien; 37, J. Garner; 22, K. Dewsbury-Hall; 42, T. Iroegbunam (booked, 22’); 18, J. Grealish; 24, C. Alcaraz (booked, 79’)(McNeil 80’); 10, I. Ndiaye; 11, T. Barry (Beto 80’).