However, during a frantic second half, Gabriel Martinelli and Hector Bellerin both scored equalisers for their team, either side of Cesar Azpilicueta putting Chelsea back in front during the closing stages.
Frank Lampard's team extend their advantage in fourth position to six points, while Arsenal remain in 10th place after a third successive draw in the top flight.
After disappointing results at the weekend, both teams started brightly, with Chelsea's Andreas Christensen squandering the first opportunity of the game by heading wide from eight yards.
Callum Hudson-Odoi then hit the crossbar for the Blues, although the winger's inswinging effort which struck the top of the woodwork appeared to be a cross rather than a shot.
Arsenal had been second best as the game headed towards the half-hour mark, and their task was made no easier when a mistake from Shkodran Mustafi led to defensive partner Luiz being dismissed.
After a short back-pass from Mustafi, Abraham was able to round Leno before setting himself to convert into an empty net, only for Luiz to cynically prevent the striker from getting his shot away.
Despite a defender getting back on the line, VAR ruled that the Brazilian deserved his red card, before Jorginho stepped up to slot the ball into Leno's left-hand corner.
While Chelsea almost doubled their lead through a fine Hudson-Odoi half-volley which was saved by Leno, Arsenal were able to see out the rest of the first half without further damage, even causing their hosts concern at the other end of the pitch.
Mikel Arteta had toyed with making an alteration immediately after Luiz's sending off, but the Gunners returned for the second half with Granit Xhaka in central defence and Mesut Ozil occupying a role in the middle of the park.
Despite the Spaniard soon feeling the need to replace Ozil, his bold approach was working with Chelsea failing to get any notable shots away in the final third.
The Blues were ultimately made to pay, although Arsenal's leveller, which came no more than 10 seconds after they had successfully defended a corner, was the home side's own doing.
As Martinelli broke forward, N'Golo Kante slipped on the halfway line to gift the youngster a way through to goal, and the forward made no mistake with a composed finish past Kepa Arrizabalaga after a 65-yard run up the pitch.
While being clawed back encouraged Chelsea to go back on the offensive, there was still a concern that Arsenal would hit them on the break, leading to the game being played end-to-end.
As both managers made substitutions, Willian tested Leno with a powerful strike after cutting inside onto his right foot, while Ross Barkley saw a glancing header well tipped behind by the German stopper.
However, Chelsea eventually got their noses back in front through Azpilicueta, who held his run perfectly to volley Hudson-Odoi's inswinging cross into the bottom corner from eight yards.
The Blues naturally would have expected to see out the rest of the contest, especially given that they remained with a man advantage despite a knock being carried by Abraham.
Arsenal had other ideas, however, as Bellerin received the ball on the edge of the area before managing to find the far bottom corner with his weaker left foot.
Lampard cut a frustrated figure on the touchline, and he could only watch on as Michy Batshuayi sent a good chance wide of the near post in the closing minute of normal time.
That proved to be Chelsea's final clear-cut opportunity, leaving Lampard with much to ponder about the development of his young team at a time when Arsenal remain on an upward trajectory under Arteta.
CHELSEA (4-3-3): Arrizabalaga; Azpilicueta, Rudiger, Christensen, Palmieri; Kante (Mount 69'), Jorginho, Kovacic (Barkley 66'); Willian (Batshuayi 79'), Abraham, Hudson-Odoi
ARSENAL (4-2-3-1): Leno; Bellerin, Mustafi, Luiz, Saka; Torreira, Xhaka; Pepe (Holding 81'), Ozil (Guendouzi 55'), Martinelli (Willock 90+1'); Lacazette