The Cherries could point to the fact that Hull managed only five shots on target, four of which went in, but they created precious little themselves and had no answer to Hull's more muscular approach.
Bournemouth had the better of the first 20 minutes but fell away once Hull got a grip of midfield and there was no way back once Ryan France had opened the scoring in the 26th minute. Robbie Stockdale crossed from the right, Nick Barmby found a gap in the defence and France scored with a low right-foot shot.
Bournemouth were rocked and five minutes later goal machine Elliott, who lines up for Northern Ireland against England next week, got the first of his two goals. Fagan brushed past two challenges on the right and Elliott got in first to score from ten yards.
Maher added to Bournemouth's problems soon after the restart when he kicked out at Fagan, an action spotted by a linesman and punished with a red card by referee Andy Penn.
Bournemouth fashioned their only real chance in the 56th minute when Matthew Mills was free at the far post to get in a blistering drive which was turned aside at full stretch by goalkeeper Boaz Myhill.
But Bournemouth's hopes of a revival were dashed a minute later when Elliott beat the offside trap to finish off a centre from Fagan.
Hull's fourth came in the 64th minute when a free-kick from Barmby was lashed in left footed by Damien Delaney and the match was over as a contest.
Substitute Brian Stock went close with a fierce long-range shot and in stoppage-time the legs of goalkeeper Neil Moss denied Hull a fifth when Jon Walters burst clear.