Elphick opened the scoring on the stroke of half-time with a thumping header before Grabban bagged his 20th of the season to keep alive Cherries’ chances of reaching the play-offs.
Armand Traore’s finely-taken goal seconds after the restart put the visitors on level terms before Harry Arter saw red for a lunge on Junior Hoilett.
Andrew Surman was restored to the starting line-up as boss Eddie Howe made one change following Cherries’ 4-2 win at Birmingham seven days ago.
Cherries striker Yann Kermorgant and QPR skipper Clint Hill both required treatment following a nasty clash of heads after just 10 seconds with the contest held up for around four minutes.
Lee Camp was the first goalkeeper called into action, the Cherries stopper diving to his left to superbly claw away a long-range effort from Jermaine Jenas after 16 minutes.
Camp then clutched a speculative strike from Benoit Assou-Ekotto before Cherries striker Lewis Grabban’s low drive was blocked by a defender.
As QPR made most of the early running, Ravel Morrison’s shot was helped over the crossbar by Camp before Matt Ritchie’s 20-yarder flashed narrowly past the upright.
Shots from Grabban and Marc Pugh were both charged down as Cherries began to find their feet and only a smart save from Rob Green prevented the hosts from taking the lead after 36 minutes.
A sweeping move down the left started by Surman ended with Green beating away Kermorgant’s first-time strike after Grabban had picked him out with a precision cross.
Grabban’s left-foot strike flew just past the post after Richard Dunne had made a dog’s dinner of trying to defend a mule-like clearance from Camp on the stroke of half-time.
Elphick nodded Cherries into the lead in first-half stoppage time when he careered through a posse of players to meet Harte’s corner with an unstoppable header from six yards.
Traore levelled just 20 seconds after the break when he neatly controlled a crossfield pass from Tom Carroll and lifted a left-foot shot past Camp from a tight angle.
Surman went close to restoring Cherries’ lead when he started and finished a flowing move, his resultant shot from just inside the 18-yard box tipped over the crossbar by Green.
Grabban turned in the loose ball to make it 2-1 after 59 minutes, the striker reacting quickly after Kermorgant’s cross from Surman’s sublime through ball had found its way to him at the far post.
Green was forced to parry Arter’s pot-shot before the Cherries midfielder received a straight red card for a tackle on Hoilett after 70 minutes.
Kermorgant was immediately sacrificed and replaced by Eunan O’Kane before Ritchie then made way with Ryan Fraser coming on after 74 minutes.
Hoilett traded passes with Austin before lifting a shot over the crossbar, while Green saved with his knees from Grabban after Fraser had sent him clear in the last minute.
Green had to tip Harte’s blistering free kick over the crossbar in the second minute of added time before Brett Pitman came off the bench to clear off the line from Dunne in the dying seconds.
Cherries: Camp, Francis, Cook, Elphick, Harte, Ritchie (Fraser, 74), Surman, Arter, Pugh (Pitman, 90), Grabban, Kermorgant (O’Kane, 70).
Unused subs: Rantie, Ward, Smith, Allsop (g/k).
Sent off: Arter.
QPR: Green, Simpson, Dunne, Hill, Assou-Ekotto, Hoilett, Jenas (Benayoun, 72), Carroll, Traore (Zamora, 80), Morrison, Maiga (Austin, 65). Unused subs: Yun, Henry, Young, Murphy (g/k).
Booked: Morrison, Dunne.
Referee: Jon Moss (England).
Attendance: 11,307 (including 1,428 away supporters).