The Blues took an early lead through a well-taken Mykhaylo Mudryk goal, but Willy Boly quickly cancelled out the Ukrainian's smart finish with a header from a free kick.
Forest continued to be the better side and went in front with 16 minutes remaining as Callum Hudson-Odoi - a former Chelsea man - curled a delightful effort into the far bottom corner.
However, Raheem Sterling followed suit at the other end before Nicolas Jackson headed the winner moments later after meeting a delightful Reece James cross.
Maximum points for Chelsea takes them level on points with sixth-placed Newcastle United, who were held to a draw by Brighton & Hove Albion earlier in the day.
As for Forest, they remain three points clear of Luton Town in the standings with one match remaining, Nuno Espirito Santo's side effectively safe unless they lose to Burnley and there is a 10-goal swing.
Chelsea fail to take advantage of fast start
Having collected seven points from matches against Aston Villa, Tottenham Hotspur and West Ham United, it should have come as no surprise when the Blues established an early lead.
Cole Palmer delivered the perfect through-ball for Mudryk to latch onto, the winger taking the ball in-field and slotting a calm finish into the far bottom corner.
However, it did not take long for Forest to get back on level terms. Morgan Gibbs-White's floated free kick to the back post found Boly, whose header deflected off Conor Gallagher and found its way into the net via the inside of the post.
That goal helped Forest stay in the ascendancy with Chris Wood causing problems for the Chelsea backline, the New Zealand international heading a good chance over as his side built some momentum.
Chelsea had a chance through Jackson at the other end, Palmer finding the run of the forward with a precise pass, but goalkeeper Matz Sels came out quickly to divert the shot behind.
Forest stunned by Chelsea comeback
A large part of the second half belonged to Forest, who struck the woodwork with a low strike from Ryan Yates soon after the restart, before the East Midlands outfit squandered a far clearer opportunity.
At the back post, Gibbs-White sent a header against the woodwork, before Wood hit the rebound - albeit under pressure from two defenders and unable to set his feet - over the crossbar from two yards.
Thiago Silva was equally frustrated having sent a glancing header from a Palmer free kick against the far post, before Hudson-Odoi became the latest player to go agonisingly close as his curling effort struck the underside of the crossbar.
However, the former Chelsea academy graduate got his reward for an effective display against his ex-employers with 16 minutes left. Much like the previous near-miss, the winger cut inside and whipped a perfect low strike into the far corner, leaving Djordje Petrovic with no chance.
Mauricio Pochettino wasted no time in sending on the calvary. Sterling had only been introduced seconds before Chelsea went behind, and Reece James followed with Gallagher making way.
Both players have either missed games through loss of form or injury, yet each made the desired impact as Chelsea turned the match around between the 80th and 82nd minutes.
Having collected the ball on the left, Sterling - much like Hudson-Odoi - drifted inside before unleashing an unstoppable inswinging strike into the far bottom corner for his seventh Premier League goal of the campaign.
With only one central midfielder on the pitch, Chelsea had little choice but to throw caution to the wind, with James and Malo Gusto working in tandem down the right.
James - on his first appearance since December 10 due to a hamstring injury - was the player who got free, keeping his composure to whip an outstanding cross on the half-volley to the waiting Jackson, who turned an overall poor display into a match-winning one with a composed header at the far post.
Although seven minutes of added-on time were indicated, Chelsea showed the kind of resilience that they have not always produced this season to get over the line, heaping the pressure on Manchester United to respond against Arsenal on Sunday in the battle for seventh position with Chelsea now holding a three-point advantage with a better goal difference.