Cherries supporters had raised £3,000 to provide free coaches for the 357 Brewers fans to make the journey to this rearranged tie and they had a shock equaliser to celebrate when Jimmy Phillips' effort deflected in to cancel out Brett Pitman's fifth-minute opener.
But Reds supporter Elphick, one of four Liverpool supporters in the Bournemouth squad, put Eddie Howe's men back in front just before the interval.
And their journey to the fourth round for the first time in nine years was eventually sealed in the final five minutes thanks to a fine finish from Ryan Fraser and Pitman's second.
The Brewers made the trip boosted by an unbeaten record stretching back to November 16 but a defence that had conceded just four goals in that 16-match run held out for just five minutes. The recalled Pitman slid in at the back post to convert Matt Ritchie's left-wing delivery for his sixth of the campaign.
Bournemouth missed chances to go further ahead through Tokelo Rantie but were made to pay for their profligacy when the visitors drew level against the run of play a minute later. Phillips was allowed time to compose himself 20 yards from goal and his speculative right-footed effort then took a hefty deflection to wrong-foot Ryan Allsop and equalise.
However, Bournemouth would deservedly head in to the break ahead. A short corner routine caught out the League Two promotion hopefuls and Elphick converted Charlie Daniels' low delivery for his first of the campaign.
Simon Francis' maintained his record of playing every minute for the Cherries this season and it was his delivery that almost provided the game's fourth goal. He provided a low ball into the path of Pitman whose low snapshot rattled off the base of Dean Lyness' left-hand post.
The 22-year-old shotstopper was keeping his side in the tie and he would make another two saves in quick succession from Pitman and then Marc Pugh from a tight angle six minutes later.
Their failure to kill off the visitors saw Eddie Howe send on top scorer Lewis Grabban, linked with a move away from the club this week. but it was fellow substitute Fraser who sealed their passage with five minutes to go when he cutely dinked the ball over Lyness from Pitman's through ball.
And the Cherries supporters could start planning for the televised fourth-round visit of Brendan Rodgers' men when Harry Arter was fouled and Pitman fired home his second of the night from the penalty spot.