Walsall seemed to be heading for their second win in four days after goals from rookie Daryl Taylor and name sake Kris put them 2-0 in front inside of an hour.
But James Hayter and Elliott struck twice in the space of two second-half minutes to turn the match on its head.
Midfielder Elliott had the first meaningful chance of the match in the eighth minute when he shot over from 20 yards after cutting in from the right.
The former Republic of Ireland midfielder Mark Kinsella was denied by a good save from Bournemouth goalkeeper Neil Moss 11 minutes before half time.
Kinsella seized on a defensive mistake from on-loan Portsmouth centre-back Eddie Howe before unleashing a fierce drive from the edge of the area, which Moss did well to keep out.
Walsall took a deserved lead just two minutes later when teenager Taylor notched up his second goal in four days.
Taylor, who marked his league debut with a late winner in Walsall's 3-2 victory over Port Vale, volleyed home from six yards out after player-manager Paul Merson had crossed from the right.
The visitors doubled their lead six minutes into the second half when defender Kris Taylor took advantage of another vintage Merson cross to head passed Moss from close range.
Elliott rattled the Walsall woodwork 18 minutes later when his 20-yard strike cannoned back off the left-hand post.
But the winger did punish Walsall soon after when his cross was smashed in at the back post by Hayter to make it 2-1.
Elliott completed Bournemouth's amazing recovery by sliding in an unlikely equalizer after substitute Steve Fletcher had headed a cross into his path.