Summer signing Elanga, introduced in first-half added time following an injury to Danilo, struck shortly after the break when he raced on to Taiwo Awoniyi’s pass and slotted home.
Nicolas Jackson missed a glorious opportunity to equalise inside the final 10 minutes, firing over from inside the six-yard box after being teed up by Raheem Sterling.
Jackson had earlier been denied by a last-ditch block by Willy Boly, while £40m deadline day signing Cole Palmer was introduced by manager Mauricio Pochettino, as Chelsea chased a response.
But despite dominating possession, territory and chances, the hosts were punished for failing to take their opportunities by a resolute Forest.
A second victory from four games lifted Steve Cooper’s visitors to ninth in the table, while Chelsea fell to 11th in the early standings.
Chelsea’s total spending in three transfer windows since Todd Boehly’s takeover surpassed £1bn following a record-breaking summer outlay of more than £380m on 10 players.
Forest, meanwhile, signed seven players during a productive deadline day – including the club-record £30m capture of PSV Eindhoven midfielder Ibrahim Sangare.
And while it was the surprise late arrival of England youth international Palmer from Manchester City which had excited the home fans, it was a promising addition for the opposition who ultimately settled the contest.
A beaming Cooper embraced Elanga following the full-time whistle as the travelling Forest fans celebrated only their second win in 21 away games since making their Premier League return last season.
Elanga, signed from boyhood club Manchester United for £15m in July, displayed speed and composure as he darted down the middle of the Chelsea defence and rolled the ball past Robert Sanchez after Moises Caicedo conceded possession.
The 21-year-old Sweden international, who had been on the pitch for a total of less than 10 minutes before he scored, proved the match-winner for Forest, for whom captain Joe Worrall – featuring despite suffering a bereavement earlier this week – and midfielder Ryan Yates were standout performers.
Singling out Worrall for praise in his post-match interview, Cooper said: “We’ve just all shown some respect to him in the dressing room. He had a brilliant ovation from the supporters too. It is an awful tragedy that he and his family have gone through.”
The Forest boss added: “To see how he has conducted himself this week, putting himself second to the team and the club – he didn’t have to do that but he has.
“Family and health is much more important than football, but I think the football has helped him this week.”
Sterling was denied by Ola Aina’s block and Ben Chilwell failed to add the finishing touch from Sterling’s superb pass in a purposeful start by the hosts, who were aiming to push on after securing a first victory of the season against promoted Luton Town last week.
Yet, despite promising build-up play, the in-form Sterling and forward Jackson were unable to inspire a fightback from the hosts.
Jackson will wonder quite how he did not equalise and set up a tense finale, but Forest, having been left empty handed from two goals up against Manchester United last week, refused to buckle under mounting pressure to clinch a significant win on their travels.
“We are disappointed because we did everything to try and get a different result,” said Pochettino. “But football is about not making mistakes and we made one mistake and paid for that.”
The Chelsea boss added: “We are building something and it is always up and down. We are unlucky because I think we deserved more. But I have said before, it is only a matter of time.”
