The former Coventry City striker raced clear of the Albion defence with four minutes to play before slotting under goalkeeper Boaz Myhill to dump the Premier League club out of the competition.
Eddie Howe's side, flying high in the Championship, dominated the first half at Dean Court as Bournemouth followed up their record 8-0 victory at Birmingham City last weekend with another impressive display to progress past the fourth round.
Eunan O'Kane's 49th-minute strike broke the deadlock, but Bournemouth had to defend resolutely to keep Albion out.
The introduction of Georgios Samaras eventually paid off for the visitors as the Greece international's effort deflected in off Tommy Elphick after 85 minutes.
However, Bournemouth immediately went down the other end and Wilson netted for the fifth game running to extend Albion winless streak to four games.
With both teams making 10 changes to their respective sides from the weekend, it was perhaps unsurprising that the opening stages were cagey, with only a Yann Kermorgant shot threatening for the hosts in the opening 10 minutes.
Adam Smith's surging run in the 14th minute created another chance for the striker, but his low shot was well blocked by a superbly-timed sliding challenge from Youssouf Mulumbu.
Midfielder Mulumbu almost opened the scoring at the other end two minutes later on a rare Albion attack, but his effort from the edge of the penalty area was easily saved by Lee Camp.
Myhill had to be alert after 20 minutes as Junior Stanislas raced through the middle, although he was unable to get a shot away and Tokelo Rantie's effort was pushed away by the Albion goalkeeper.
Brown Ideye was given his first sight of goal just after the half hour after Sebastian Blanco and Victor Anichebe combined to play him in, with Camp equal to the Nigerian's weak effort on goal.
Bournemouth finally got reward for their efforts four minutes into the second half through O'Kane.
Another surging Stanislas run put the Albion defence on the back foot and his touch inside was met by the midfielder to calmly slot home his first goal of the season.
Albion felt they should have been level minutes later when Ideye's cross appeared to be handled by Elphick inside the penalty area, but their protests were waved away by referee Paul Tierney.
Stephane Sessegnon's introduction from the bench almost brought the visitors back into the game as he fired a 25-yard shot just off target.
Albion did equalise with five minutes to play, albeit with the aid of a deflection, as Samaras' shot flicked off Elphick and past Camp.
The scores were only level for 50 seconds, though, as substitute Wilson raced clear on the right and found a way past Myhill.
Chris Baird then blazed a header over the crossbar from Saido Berahino's cross as Albion spurned a late chance to force extra time.
