Cesc Fabregas had earlier missed from the penalty spot for Chelsea on what is expected to be his final appearance for the club before departing for Monaco, but the hosts cruised over the line in the end to progress from round three for the 21st season in a row.
Reverting to his second-string side for this first cup outing of 2019, Maurizio Sarri somehow saw his side go into the interval all square despite a dominant first-half showing.
Morata was guilty of missing a huge chance when heading Zappacosta's delightful cross right at Luke Steele, with plenty of the goal to aim for either side of Forest's back-up keeper.
Chelsea lost a bit of their momentum late in the first period when Loftus-Cheek hobbled off with a minor injury, leading to the introduction of Eden Hazard for the remaining 48 minutes.
Forest, fortunate not to have makeshift centre-back Claudio Yacob dismissed for a last-man trip on Morata in the first half, held out for four minutes of the second period before conceding.
Following some good wing play from Bayern Munich target Hudson-Odoi, Morata got between the two Forest centre-backs and converted the youngster's cross, leaving Steele with no chance of keeping this one out.
Hudson-Odoi sparkled on his third start under Sarri, shifting the ball away from his man and firing a shot on target, moments before Daryl Murphy headed the ball right at Willy Caballero at the other end.
The game was put out of Forest's reach an hour into the match thanks to a second for Morata who, moments after guiding Ross Barkley's cross over an unguarded net from three yards out, glanced Hudson-Odoi's delivery into the far corner to redeem himself.
Barkley and Hudson-Odoi both had shots kept out by Steele in the remaining minutes, but the biggest cheer was reserved for Fabregas, who departed the Stamford Bridge field for one final time late on.
CHELSEA (4-3-3): Caballero; Zappacosta, Christensen, David Luiz, Emerson; Fabregas (Kante 85'), Ampadu, Barkley; Loftus-Cheek (Hazard 42'), Morata (Azpilicueta 74'), Hudson-Odoi
NOTTINGHAM FOREST (4-2-3-1): Steele; Darikwa, Yacob, Fox, Janko; Guedioura, Colback; Cash (Lolley 57'), Carvalho, Osborn; Murphy