Despite barely getting out of their own half for large parts of the second half, Chelsea just about got the job done in normal time through that late Morata strike to remain on course for an eighth success in this competition.
The Cherries, aiming to make it through to their first semi-final of a major competition, allowed their opponents to enjoy plenty of the ball in the first half.
A wayward Gosling shot aside early on, it was backs-to-the-wall stuff and they were behind with 13 minutes played in West London.
Having been made to remain patient in search of an opener, Kenedy injected a bit of quality on his third start of the season by back-heeling the ball into the path of Cesc Fabregas, who in turn picked out Willian for a simple finish.
The replacement attacker wasted Bournemouth's first real chance of the game, shanking the ball into the stands when pouncing on a Willian error, while Lys Mousset could only pick out the hands of Willy Caballero with his downward header.
Chelsea continued to see more of the ball, coming close to a second when Gary Cahill belted the ball over the crossbar after Steve Cook and Simon Francis put their bodies in the way to block a couple of attempts.
Bournemouth, who had failed to score in nine of their 21 matches prior to tonight, saw the momentum build in their favour for a large portion of the second half.
Turning possession into chances was proving to be a problem, though, with the Cherries seeing four of their attempts - the best coming via Mousset after good work from Ibe - blocked behind for a corner prior to the hour mark.
Gosling skewed another effort horribly wide, but Bournemouth's dominance was worrying Antonio Conte and the Italian turned to his bench - Eden Hazard, Tiemoue Bakayoko and matchwinner Morata all coming on in the space of 12 minutes.
Chelsea's fortunes did not improve a great deal, however, as it was still the visitors asking all the questions as they searched for just a second win in 10 attempts against tonight's opponents.
Ibe shot wide following a smart touch from substitute Callum Wilson to set him up, and then Wilson himself came close by angling in a shot on the spin for Caballero to turn aside.
Bournemouth would get the goal that their determination deserved, courtesy of a fine curled effort from Gosling right on the edge of the box that left Caballero with no chance.
Eddie Howe's men barely had a chance to compose themselves before Morata snatched the tie away from them, however, as the Spaniard exchanged passes with Hazard and tucked the ball through Artur Boruc's legs.
Chelsea: Caballero; Rudiger, Ampadu, Cahill; Zappacosta, Drinkwater, Fabregas, Kenedy; Willian (Hazard 61'), Batshuayi (Morata 73'), Pedro (Bakayoko 61')
Bournemouth: Boruc; A.Smith, Francis, S.Cook, Simpson; Surman, Arter (L.Cook 74'), Gosling; Mousset (Wilson 71'), Fraser, Defoe (Ibe 17')