But Ivory Coast right-back Aurier levelled after the break and Chelsea were fortunate to escape without an embarrassing defeat after Forest dominated for long spells in the second half.
Chelsea are languishing in eighth place after failing to win for the sixth time in their last seven league games.
The Blues are seven points adrift of fourth-placed Manchester United in the race to qualify for the Champions League via a top-four finish.
Just months after replacing the sacked Thomas Tuchel, Chelsea boss Graham Potter is starting to come under fire as the early optimism after his move from Brighton begins to fade.
Chelsea's 2-0 win against Bournemouth on Boxing Day was supposed to offer hope of a brighter period in 2023.
But while it is surely too early for Potter to be worried about losing the first high-pressure job of his career, the 47-year-old needs an uptick in results in the new year to avoid putting Chelsea chairman Todd Boehly in an awkward position.
Forest are third bottom having won just three games all season, but boss Steve Cooper will take heart from the way his team rattled Chelsea in the second half.
The only dark cloud for Forest was the alleged homophobic chanting from home supporters that earned condemnation from LGBTQ groups.
Brennan Johnson threatened an early goal for Forest when he sprinted clear of Marc Cucurella and Thiago Silva, but Kepa Arrizabalaga sparred his teammates' blushes with a good save.
If that was a lucky break for Chelsea, Potter's side enjoyed an even greater stroke of good fortune as they took the lead in the 16th minute.
Christian Pulisic's cross towards Kai Havertz was deflected by Forest defender Willy Boly's out-stretched foot, with the ball looping onto the crossbar and bouncing down perfectly for the unmarked Sterling to slam home from close range.