Scotland winger Fraser completed a slick counter-attack to put the hosts ahead in the first half and then struck the winner with a deflected effort in the last minute.
Idrissa Gueye’s second-half equaliser, assisted by Oumar Niasse just three minutes after he came on, gave Allardyce hope of becoming the only Everton manager to go unbeaten in his first seven Premier League games.
But even Everton goalkeeper Jordan Pickford, who denied Callum Wilson and Jordon Ibe in each half, could not stop Fraser’s effort as it flew in off defender Michael Keane’s outstretched leg.
The win moves Eddie Howe’s Bournemouth five places up the table to 13th and means they avoid going nine league games without a win, although they lost striker Joshua King to a possible hamstring injury in the first half.
Everton stay ninth, level on 27 points with Leicester, but conceded more than one goal in a game for the first time since their 4-1 defeat at Southampton on 26 November.
