Willian had given Chelsea a 13th-minute lead but the Blues invited pressure and Dan Gosling equalised in the 90th minute.
Chelsea attacked from the restart and Eden Hazard back-heeled into the path of fellow substitute Morata, who tucked in to send the Blues through.
The much-changed home side struggled to assert themselves against the Cherries but eventually prevailed, extending the visitors’ winless run to seven games ahead of Saturday’s trip to runaway Premier League leaders Manchester City.
Head coach Antonio Conte has added a dose of realism to Chelsea’s bid to retain the Premier League title with the Blues third, 14 points behind City, but the quest for silverware this season remains alive and well.
Ethan Ampadu started as Conte, as promised, rotated his options making eight changes. The Cherries made seven following the 4-0 loss to Liverpool.
Ampadu began with a rush of blood but otherwise performed admirably in a testing contest. The 17-year-old was booked for a foul on Jermain Defoe, who at 35 is more than twice the Chelsea defender’s age, in the second minute.
Defoe required treatment and continued for 15 minutes before departing after an apparent apology from Ampadu. Bournemouth were a goal down by then.
It was a fine team effort as Michy Batshuayi turned to feed Kenedy, whose back heel sent Cesc Fabregas through. The Spain playmaker might have shot himself but showed composure to centre for Willian to make a simple tap-in.
Willian stopped expecting a foul when Harry Arter stole the ball from him and prodded it to Jordon Ibe. Defoe’s replacement sliced a shot wide.
Pedro’s pace created a chance for Batshuayi but the striker’s first touch allowed Steve Cook to make a covering block.
Batshuayi fell meekly clutching his face under the slightest attention from Adam Smith and Simon Francis, with referee Lee Mason deeming the incident not even worthy of a shake of the head.
Bournemouth pressed forwards in the second half, with Chelsea struggling to alleviate the pressure.
Conte did not seem impressed, even less so with Willian, who for a third time in the match stopped playing wanting a foul.
This time it was close to the Chelsea box. Conte had a word with his bench and moments later Eden Hazard was introduced for the Brazilian.
Tiemoue Bakayoko came on for Pedro and later Morata for Batshuayi, as Conte altered his 3-4-3 formation to 3-5-2.
Conte might have preferred to keep them on the bench ahead of Saturday’s Premier League clash at Everton and use teenagers Dujon Sterling or Callum Hudson-Odoi instead.
But Bournemouth’s pressure was in danger of telling, and it finally did when Morata’s weak clearing header was played by Ibe to Gosling, who curled in from the edge of the area. Chelsea’s lead was restored almost immediately as Davide Zappacosta’s hopeful pass was flicked by Hazard into Morata’s path and he squeezed the ball under goalkeeper Artur Boruc.
His celebration brought a yellow card which means the striker is suspended for Saturday’s match at Goodison Park.
Caballero, Rudiger, Ampadu, Cahill, Zappacosta, Drinkwater, Fabregas, Kenedy, Willian (Hazard 61), Batshuayi (Morata 73), Pedro (Bakayoko 61).
Boruc, Adam Smith, Francis, Steve Cook, Simpson, Fraser, Arter (Lewis Cook 74), Surman, Gosling, Defoe (Ibe 17), Mousset (Callum Wilson 71).