After the first leg ended goalless a fortnight ago, the two London rivals met for the fifth time this season and the third time in January alone, with each previous contest ending all square.
Arsenal finally ended the season-long stalemate between the clubs, though, with Granit Xhaka the unlikely scorer of the winner after an Antonio Rudiger own goal had cancelled out Eden Hazard's opener in an action-packed start to the match.
The Gunners - who have never won the competition under Arsene Wenger - will now take on Manchester City in the final at Wembley on February 25.
Chelsea took just six minutes to race into a two-goal lead during their weekend win over Brighton & Hove Albion, and they thought that they had made another quick start when Pedro nodded Cesar Azpilicueta's cross past David Ospina after only five minutes, only to be correctly flagged for offside.
Arsenal were also fast starters at the weekend with four goals in the opening quarter of the game, but it was the opposite this evening and Wenger's side were punished for their early sloppiness after only seven minutes when Hazard latched on to Pedro's through-ball before burying his finish into the bottom corner with aplomb.
That goal suddenly sparked the hosts into life, though, and they almost equalised just two minutes later when Jack Wilshere broke into the box from Alex Iwobi's pass before drawing a save from Willy Caballero, who recovered in time to push the follow-up away from the advancing Nacho Monreal.
Monreal would have a decisive say in the equaliser moments later as he continued his purple patch in the final third following a goal and two assists against Crystal Palace on Saturday, losing his marker from a corner and seeing his header deflect off Marcos Alonso and then the unfortunate Rudiger before bouncing over the line for an own goal.
The two sides had played out one thrilling 2-2 draw at the Emirates Stadium and two not-so-entertaining goalless stalemates at Stamford Bridge already this season, and the start of tonight's game suggested that it would firmly sit in the former category.
Chelsea quickly regained control of the game, though, and a clever dummy from Hazard provided Willian with the chance to restore the visitors' lead in the 20th minute, only for the Brazilian to drag his left-footed strike wide.
Willian would not last much longer either, limping off just before the half-hour mark due to a hamstring injury as Ross Barkley came on for his Chelsea debut and first appearance of the season.
Arsenal's only period on top during the first half came during the five minutes immediately after the opening goal, yet it was the Gunners who came closest to taking the lead into half time.
Xhaka saw one ambitious free kick from a tight angle deflect off Hazard in the wall and ripple the top of the net on its way over, before Mesut Ozil almost nicked the lead in the final minute of first-half stoppage time with another deflected drive which bobbled inches wide.
The Gunners started the second half on top too, and they were rewarded for their substantial improvement on the hour mark when they deservedly took the lead.
Alexandre Lacazette's low cross into the box deflected off the hapless Rudiger and into the path of Xhaka, who reacted quickly to stab his finish into the far corner.
Chelsea nearly responded immediately when Pedro's chip was plucked out of the air by Ospina, but the visitors were a shadow of the team that controlled the first half and Arsenal remained the dominant force in the match after taking the lead.
Indeed, the hosts should have killed the tie off with just 14 minutes remaining when a sweeping team move ended with Ozil playing the ball into the path of Iwobi, but with only Caballero to beat the Arsenal man fired his finish too close to the keeper.
Rudiger had the chance to punish that profligacy and atone for his own mishappenings from a dangerous Barkley corner in the 82nd minute, but with Ospina in no-man's land the Chelsea defender could not get a clean connection on his header as it looped harmlessly over the crossbar from close range.
Such chances were few and far between even in the closing stages, though, as Arsenal held on to seal their place in next month's final, where they will face quadruple-chasing Man City.
It is only the third time that they have reached the final of this competition under Wenger, who has the chance to clinch the only major domestic silverware which has eluded him in England so far, having previously lost the 2007 and 2011 finals.
The Gunners are not in action again until next Tuesday when they take on bottom-of-the-table Swansea City, whereas Chelsea - whose 12-match unbeaten streak comes to an end - will host Newcastle United in the FA Cup fourth round on Sunday.
ARSENAL (4-3-3): Ospina; Bellerin, Mustafi, Koscielny, Monreal; Elneny, Xhaka, Wilshere; Iwobi (Ramsey, 84'), Ozil, Lacazette (Kolasinac, 84')
CHELSEA (3-4-3): Caballero; Azpilicueta, Christensen, Rudiger; Moses (Zappacosta, 72'), Bakayoko, Kante, Alonso; Willian (Barkley, 30'), Hazard, Pedro (Batshuayi, 65')