The scoring started after just seven minutes when Wayne Routledge picked out Liam Cullen for his first of the game.
Grimes doubled the home side's lead after a sensational strike found the top corner.
10 minutes into the second half, Forest were given a lifeline after Anthony Knockaert fired home from close range.
The lifeline was short-lived after Ryan Manning won a penalty which was emphatically converted by Grimes.
Just minutes later Manning found youngster Cullen who fired home past Jordan Smith.
Swansea made it five when Oliver Cooper fired goalwards and the ball squirmed under the body of Smith.
Story of the match
Both sides rotated and made multiple changes. Swansea made eight changes with Conor Hourihane making his debut, meanwhile Forest made 10 with captain Joe Worrall the only player to keep his place in the starting 11.
The first real chance of the game resulted in a goal for Swansea. In just the seventh minute, a great ball from Routledge found an unmarked Cullen who volleyed home past the helpless Smith.
After getting his assist, Routledge picked an injury and was replaced by the in from Jamal Lowe after just 20 minutes.
Just two minutes after coming on, Lowe was played through after a delightful flick from Cullen, but his touch was too heavy and the ball was collected by Smith.
The first chance for Forest fell to Joe Lolley who reached the byline and tried to pull the ball back to the incoming Knockaert, but the pass was well intercepted by Marc Guehi.
Swansea doubled their lead in superb fashion. Matt Grimes picked up the ball from 30 yards and his thunderous left-footed strike found the top right-hand corner leaving Smith with no chance.
The home side thought they made it three Connor Roberts poked the ball home, but his joy was short-lived after the full-back was denied by the offside flag.
Forest had an awful first half and failed to create any clear cut chances, not evening registering a single shot on target. Knockaert's speculative effort in just the fourth minute was as good as it got for the Reds.
In a bid to get back into the game, Forest brought on Cafu and Miguel Guerrero in place of Luke Freeman and Fouad Bachirou in what was their first game back from injury.
Hughton's side pounced on a rare error from the Swansea defence and Lolley played the ball to Knockaert and the Frenchman fired past Freddie Woodman from 12 yards out to halve the deficit.
Minutes after getting back into the game, the Swans’ two-goal lead was restored after Worrall was controversially adjudged to have brought down Manning in the area. Captain Grimes stepped up and sent Smith the wrong way.
Two goals in just five second minutes put the game beyond Forest. Manning found space down the left-hand side and drilled the ball low into the box to find an unmarked Cullen for his second of the game.
Five minutes from time, Tobias Figueiredo gave the ball away to Lowe who drove towards the Forest goal before his deflected to effort fell to substitute Cooper who fired his shot under the body of Smith who really should have done better.
Man of the match: Matt Grimes (Swansea City)
The midfielder led by example and helped his side to a comfortable victory with two goals.
His first goal could be a contender for goal of the round after firing home from 30 yards out. The Swans number eight then showed great composure to score his penalty and put the game beyond Forest.
The midfielder barely made a mistake and was well worth the Man of the match award.