Taiwo Awoniyi headed Forest in front early at Stamford Bridge, nodding home from Bilal Dakwa's cross before Igor Jesus converted a penalty after Malo Gusto pulled Awoniyi's shirt in the box.
The result moved Nottingham Forest six points above the relegation zone and maintained the gap over fifth-placed Aston Villa ahead of a Europa League semi-final tie that will determine whether the club reaches the final in Istanbul on 22 May.
Gibbs-White had come off the bench in the second half and assisted before colliding heavily with Sanchez, walking off the pitch clutching his face. Pereira had planned to rest him ahead of the Villa Park second leg and was forced to withdraw him after just 20 minutes on the pitch. The manager refused to rule him out for Thursday.
"I'm waiting. I am not a doctor. Gibbs-White has a deep cut. We will see. He is a fighter and he has a strong mentality. I hope our medical department can do magic to put him in condition to play against Aston Villa on Thursday. I said to him — you don't need to head the ball just to play."
Joao Pedro struck a fine late consolation for Chelsea from the edge of the box, but it was not enough to prevent a sixth consecutive Premier League defeat for the Blues, who have now dropped to sixth place and face an anxious wait over their European qualification. The result kept Nottingham Forest inside the top four on goal difference.
Pereira reserved considerable praise for his squad's collective identity, describing them as one of the best groups he had worked with throughout his career.
"The only thing I'm sad about is the injury of Gibbs-White. The speed of the team — we can change the players, but the speed will be there. We are a team that want to keep our future in our hands and to do that we need to win games and not be worried about the results of other teams. If we win, we keep the distance and today another game, they proved again the speed, the quality and organisation."
On the specific tactical challenge Chelsea presented by loading the centre of the pitch, Pereira identified how Forest overcame it.
"One thing we are doing well is our pressure, we are organised. When we play these teams that put a lot of players in the middle, it is important to close the space inside. We have players that can score goals in different ways."
He spoke about Awoniyi's contribution beyond goalscoring, having fielded him despite the striker not being registered for the Europa League.
"Taiwo is a danger player. He is not on the list in the Europa League but we know he will push the defensive line of the opponents because he runs, attacks the space, holds the ball — and this is the kind of spirit and energy we need."
He closed with a message of confidence ahead of Thursday's second leg, Forest trailing Aston Villa 0-1 from the first leg.
"I'm sure we will go to play a very difficult game. We go with energy, spirit and fight to the last minute. I'm sure about that — the players we have for the first XI, and also the players on the bench, will come with fire to help the team."