The visitors were tidy on the ball but could not cope with Chelsea's attack, and the gulf in class showed on the board in the 13th minute.
The Blues doubled their lead five minutes later as Hazard's deflected through pass fell for Batshuayi, who poked through the legs of the Forest goalkeeper.
Forest came within inches of pulling one back with their best effort of the match, but Kieran Dowell's 20-yard free kick came back off the crossbar.
Chelsea killed the tie before half time as Musonda marked his first senior start for the club with a goal. Cesc Fabregas picked him out with an intelligent reverse pass, and the 20-year-old thumped a shot through Henderson, who could be adjudged culpable for two of the first three goals.
To make matters worse for the Tricky Trees, Hazard began to really influence the game in the second period and, minutes after hitting the woodwork with a curling strike, he set up Batshuayi with some magical build-up play.
The Belgian latched onto another fine Fabregas find and skipped away from two physical Forest challenges before rounding Henderson, only to turn his back on the vacant goal and square for Batshuayi to have an easy finish.
Chelsea's stand-in striker knew nothing about his third on 84 minutes. Kenedy rattled the crossbar with a thunderous shot from the left edge of the area, and the ball rebounded off Batshuayi and into the unguarded net.
Conte used his side's comfortable cushion to give 17-year-olds Ethan Ampadu and Dujon Sterling debuts from the bench, but it was a Forest substitute who completed the scoring to give the visiting fans something to cheer.
The Reds broke down the right and advanced into the Chelsea box with some nice passing, although there was an element of fortune as the ball fell to Tendai Darikwa to grab their consolation with the last kick of the game.