The impressive Fraser opened the scoring just after the half-hour mark. A soft goal from an Everton point of view as James McCarthy gave the ball away cheaply and allowed the hosts to break.
Good play out wide from Josh King meant he could angle a cross towards Fraser and the winger volleyed home as he found himself unmarked from 10-yards out inside the Everton penalty area.
The hosts repaid the favour just before the hour mark when Steve Cook gave the ball away just outside the Bournemouth 18-yard-box to Gylfi Sigurdsson.
Sigurdsson picked out Oumar Niasse and he played a reverse ball to his fellow countryman Idrissa Gueye and the midfielder was able to place the ball past a beaten Asmir Begovic to draw the Blues level.
Yet Fraser wrapped the game up with just a handful of minutes to go.
The 23-year-old beat three Everton defenders before striking an effort just inside the 18-yard box. The effort took a wicked deflection and looped over Jordan Pickford to hand the hosts all three points.
Away day blues
Everton have struggled away from home for the best part of 12 months and the first-half at the Vitality Stadium was no different.
With one eye towards Monday’s New Years’ Day tie with Manchester United, Sam Allardyce opted to make four changes with Yannick Bolasie, Tom Davies, Mason Holgate and Ashley Williams all dropping to the bench.
McCarthy was making his first Premier League appearance since last March and it showed as he struggled to get a foothold in the game.
His misplaced pass that allowed the Cherries to break away and take the lead was exhibit A of just that rustiness. With Cuco Martina down on the floor, the midfielder had the chance to put the ball out of play.
He didn’t and it cost the Blues dearly.
In attack, it was the same old story for Allardyce’ side. Begovic could have taken a quick trip to Bournemouth beach for the first half hour as he didn’t touch the ball for the best part of 40 minutes.
The 63-year-old Everton boss is targeting a striker in January so that Dominic Calvert-Lewin does not burn out too quickly but if he doesn’t act fast, that may be too late.
With no other real options, Calvert-Lewin continued his away day to an island of isolation between opposition defenders - with yet to no support from his fellow attackers.
Aaron Lennon attempted his best, he created two chances for the Blues, but it wasn’t enough. For another game, club record signing Sigurdsson failed to make a real impact in the forward areas.
Niasse impact but Blues falter
Just like they did when the sides met earlier in the campaign, Everton needed Oumar Niasse to bail them out.
His energy replaced a tired Calvert-Lewin and gave the visitors an option.
He had an instant impact when he set up Gueye for the equaliser with his first touch on the pitch but ultimately, the visiting Toffees would falter.
Allardyce's side had chances to snatch the three points when Bolasie blazed a chance over the crossbar and then Sigurdsson was denied brilliantly by Nathan Ake but the day would belong to Fraser.
The young Scotsman was the outstanding player on the pitch from the first whistle and he made it count right before the final whistle, lifting Eddie Howe's side to 13th in the table.