England internationals Callum Hudson-Odoi and Ross Barkley both found the target for the Premier League side but Forest produced a spirited display and were unlucky not to take more from the game had decisions gone their way.
Marginal calls from the video assistant referee saw a decision to award Forest a first-half penalty following a foul on Alex Mighten reversed and a second-half goal by Ryan Yates chalked off for offside.
Sabri Lamouchi made a host of changes for the trip to Stamford Bridge, with Jordan Smith making his first start of the season and Alex Mighten making his full debut, while Carl Jenkinson returned to the side after a lengthy period out through injury.
It was also a proud day for the club's academy, with five graduates named in the starting line-up and a total of 10 in the 18-man matchday squad.
Chelsea, who also made several changes to their starting 11, got off to the perfect start. Callum Hudson-Odoi got the early breakthrough after cutting in from the right-hand-side and firing a shot low and hard past Smith.
Despite the early set back, Forest remained in the match and thought they had the perfect opportunity to draw level when they were awarded a penalty on 21 minutes. Mighten was played through by Albert Adomah and tripped in the area by Fikayo Tomori, however after further inspection by VAR the penalty was ruled out after Mighten was adjudged to be in an offside position in the build up.

The home side were quick to reassert their authority on the match by making it two just after the half-hour mark. Hudson-Odoi again cut in from the right and unleashed a powerful shot which Smith could only parry straight into the path of Ross Barkley. The former Everton man tapped in from close range to double Chelsea's lead.
Mighten and Johnson continued to look lively for the visitors on the counter attack but Forest were unable to find a reply to Chelsea's two-goal lead at the break.
Yohan Benalouane replaced Tobias Figueiredo at the heart of the Forest defence at the interval and it was a cagey start to the second half with neither side particularly dominating possession. Joao Carvalho did test Willy Caballero from the edge of the host's penalty box, but the Portuguese's effort was straight at the Chelsea stopper.
The visitors did find a way past the Argentine on 66 minutes but once again VAR came to the rescue of the home side. Carvalho's cross was nodded in by Ryan Yates, who had found space in the penalty area, however the midfielder was again adjudged to have been in an offside position.

Tyrese Fornah and Yassine En-Neyah were then introduced into the match, replacing Alfa Semedo and Brennan Johnson, to make their first-team debuts after graduating through the Forest academy.
Despite being two down, Forest were playing with confidence and looked threatening on the break. Mighten was a constant threat for the Chelsea back four and after he was brought down after being through on goal, The Reds again came inches from getting a goal back. Carvalho's cross was poked towards goal by En-Neyah but unfortunately the substitute was unable to get enough of a touch on the cross to trouble Caballero.
A defeat and an end to Forest's FA Cup journey this season but there are plenty of positives to take from the game as The Reds return their focus to the Championship.
Forest: Smith, Jenkinson, Dawson (c), Figueiredo (Benalouane 46'), Ribeiro, Yates, Semedo (Fornah 69'), Carvalho, Johnson (En-Neyah 81'), Adomah, Mighten
Unused subs: Shelvey, Sole, Hefele, Gabriel
Booking: Yates 77'
Chelsea: Caballero, James, Christensen, Tomori, Emerson, Barkley, Jorginho (c), Kovacic (Mount 69'), Hudson-Odoi, Pedro (Lamptey 75'), Batshuayi
Unused subs: Cumming, Rudiger, Willian, Giroud, Azpilicueta
Goals: Hudson-Odoi 6', Barkley 33'
Booking: James 65'
Referee: Peter Bankes
Attendance: 40,492 (2,788 away)