The Seagulls had their moments in the match too, making 11 shots in total, but lacked the cutting edge to finish them off.
Then, in the dying moments, substitute Oliver Giroud fired a header straight at Sanchez from a free-kick conceded by Ben White, who brought down Callum Hudson-Odoi and got sent off for that.
Despite the draw, Thomas Tuchel's side moved up to fourth position, but only by the courtesy of a superior goal-difference.
Player ratings for Chelsea
Kepa Arrizabalaga - 7/10
With the Seagulls laying siege to Chelsea's half, Kepa had to be on his toes all the time, despite making just two saves on the night. He was beaten by Danny Welbeck's late effort, but luckily for him, the woodwork came to his aid.
Andreas Christensen - 6/10
Christensen returned to Tuchel's XI after missing last time out with an injury and looked at home on the right-side of the back three. He coped well with the threat of Leandro Trossard.
Kurt Zouma - 7/10
Brighton pressed him a lot but Zouma stood firm, though he conceded a huge chance in the second-half which the visitors ended up wasting. The Frenchman also had a shot at goal early on, but Sanchez was on hand to save it.
Antonio Rudiger - 7/10
The German defender has been excellent under Tuchel and turned in another solid shift at the back. He wasn't afraid to step into midfield and made one crucial block to keep out Welbeck late in the game.
Reece James - 6/10
It wasn't the best night for the 21-year old, who couldn't influence proceedings going forward. He struggled to link up with Pulisic and didn't look comfortable after switching to a left-wing-back role after Alonso was subbed on.
Jorginho - 6/10
He was elegant and tidy with the ball, like always, whilst also making a vital block to deny Yves Bissouma, whose goal-bound effort was turned into the roof of the net. However, in the second-half, Jorginho struggled to break down Brighton's defense.
Mason Mount - 6/10
Mount was energetic and moved the ball quickly, even playing a couple of nice passes forward, but operating from a deeper role limited his influence in the game.
Marcos Alonso - 6/10
Having lost his starting place to Ben Chilwell, the Spaniard was under pressure to deliver tonight, but failed. He didn't get forward much and Brighton had little problems while dealing with him.
Hakim Ziyech - 6/10
Chelsea's hero in the FA Cup semi-finals on Saturday, Ziyech, cut a listless figure tonight. He conceded possession cheaply at times and was never really an attacking menace.
Kai Havertz - 6/10
A tame first-half effort set the tone for the rest of his outing as Havertz failed to impress too, never again testing Brighton in the match.
Christian Pulisic - 6/10
The American has been in good form since returning from international duty but tonight he was anything but. Pulisic was kept at bay by Brighton's well-drilled backline and saw a late effort saved too.
Olivier Giroud - 6/10
He had two late chances, but fired the first one wide off target and headed the next straight at Sanchez.
Callum Hudson-Odoi - 6/10
He won a free-kick late on but other than that, didn't have much to show for his efforts.
Timo Werner - 5/10
Tuchel subbed him on to try and find a late winner but Werner was barely involved.