Antonio Conte's side dominated possession for long periods at Stamford Bridge -- where the Video Assistant Referee system (VAR) was in operation -- but lacked the killer instinct required to take a step towards the final.
They had 21 shots to just eight from Arsenal, yet the tie remains on a knife-edge heading into the second leg at the Emirates Stadium on January 24.
While it was a frustrating evening for Chelsea as they chase a first League Cup final berth since 2015, the stalemate was a welcome tonic for their London rivals.
Arsenal, who also drew both Premier League encounters with Chelsea this season, suffered one of the most chastening defeats of Arsene Wenger's reign on Sunday as second tier Nottingham Forest won 4-2 in the FA Cup third round.
While Arsenal were well below their best once again, in contrast to their surrender in Nottingham, the Frenchman will be encouraged that they showed enough spirit to keep Chelsea at bay.
Conte had mostly fielded fringe players en route to the last four, but Eden Hazard, back from a calf injury, N'Golo Kante and Thibaut Courtois were called upon with a Wembley showdown against Manchester City or Bristol City in sight.
Wenger surprisingly left Alexis Sanchez on the bench amid talk of the unsettled Chile forward joining Manchester City, while Mesut Ozil missed out with a knee injury.
Wenger was watching from the press box as he serves his touchline ban and the Gunners boss was shifting anxiously in his seat early on when Hazard's pass gave Alvaro Morata an opportunity that he drove into the side-netting.
After a tepid opening from Arsenal, Alexandre Lacazette escaped the Chelsea defence to collect Jack Wilshere's pass, but the French striker wasted the chance as he scuffed a hurried shot wide.
Chelsea wing-back Victor Moses forced David Ospina into action with a stinging drive that the Arsenal goalkeeper fumbled before recovering to snatch the ball away from Morata.
- Ineffective -
Moses was proving a problem for young left-back Ainsley Maitland-Niles and the Nigerian found space for another shot moments later, this time his low shot hit Ospina's near post before Arsenal scrambled it away.
Alex Iwobi was selected despite Wenger saying he could be fined for reportedly partying until the early hours of Saturday morning and the Arsenal winger almost repaid his manager's faith with a drive that Courtois pushed away.
When Maitland-Niles went down under pressure from Moses in the area, referee Martin Atkinson consulted the Video Assistant Referee, but no penalty was given.
Fabregas should have given Chelsea the lead just before half-time, but the former Arsenal midfielder wastefully headed straight at Ospina from close-range.
Andreas Christensen fared no better after the interval when the Chelsea defender dived to head over from inside the six-yard box.
Chelsea were piecing together their most convincing spell of pressure, but they couldn't find the finishing touch as Morata's powerful effort was well saved by Ospina.
Wenger's response was to send on Sanchez for the ineffective Lacazette with 25 minutes left, yet within seconds Moses almost broke the deadlock when his goal-bound shot was deflected over by Shkodran Mustafi.
When Christensen headed wide of an open goal after Ospina misjudged a corner, Conte must have feared it wouldn't be Chelsea's night.
So it proved as Willian was denied by Ospina and Fabregas clipped a post in the closing moments.