Wigan - who were only denied victory in the initial tie by a late equaliser - will now take on West Ham United in the fourth round.
The hosts began at a good tempo and won possession in a dangerous position on a couple of occasions in the opening exchanges, with one of those leading to Gavin Massey drawing the first save of the game from Artur Boruc.
Boruc was called into action again moments later when he denied Will Grigg's flick with a smart reaction save, but there was nothing the Bournemouth keeper could do about the rebound as Morsy slid the ball home to give the League One side a shock early lead.
The Cherries were soon on the search for an equaliser, though, and Ryan Fraser came close to providing it when he fired a free kick wide of the target after 15 minutes.
The pressure continued to grow from the Premier League side in an increasingly feisty affair, but Wigan protected their goalkeeper well until Jamie Jones was forced to make a low stop to keep out Benik Afobe's flicked header shortly after the half-hour mark.
That was about as good as it got for Bournemouth in the first half, but within four minutes of the restart they were threatening again when Jones pushed Fraser's low strike behind for a corner.
Wigan turned to Noel Hunt off the bench in an effort to change the rhythm of the match, and the switch almost had an immediate impact when the substitute teed up Gary Roberts, who bent his effort wide of the left post.
Max Power then sent a teasing ball into the box which no-one could get on the end of before Ryan Colclough held off a Bournemouth defender and flashed an effort just past the upright during a resurgent spell for the Latics.
Eddie Howe's side soon regained a degree of control over the match, but the equalising goal was still eluding them as both Afobe and Lys Mousset failed to hit the target after being picked out by crosses into the box.
Afobe had another chance to level things up with a little over 20 minutes remaining when he collected Jordon Ibe's pass in the right channel only to fire his subsequent effort inches wide of the far post.
Having weathered the Bournemouth storm Wigan began to fight back in the final 20 minutes, though, with Massey seeing an effort deflect narrowly wide following good work from Burn.
Burn would go one better from the resulting corner as he doubled the home side's lead against all the odds, turning the ball in off the underside of the crossbar after the delivery had bounced all the way through to him.
Suddenly Wigan were in dreamland having been hanging on for much of the second half, and things got even better just three minutes later when another Power delivery yielded a third goal, this time Elder looping his header past the keeper for his first Wigan goal.
Bournemouth could only muster an Ibe effort which crashed into the side-netting as a response, with League One side Wigan holding out to inflict a first defeat in six games on their visitors.
Wigan are now unbeaten in 13 games themselves and will come up against Premier League opposition once again in the next round when they face West Ham.
WIGAN (4-2-3-1): Jones; Power, Dunkley, Bruce, Elder; Perkins, Morsy, Colclough (Burn, 71'), Roberts (Jacobs, 86'), Massey; Grigg (Hunt, 51')
BOURNEMOUTH (4-4-1-1): Boruc; Fraser (Hyndman, 78'), Cook, Simpson, Daniels; Mahoney (Ibe, 64'), Surman (Taylor, 87'), Arter, Pugh; Mousset; Afobe