Pep Guardiola's City reached the Wembley showpiece the previous evening by beating Championship side Bristol City 3-2 in the second leg of their tie and 5-3 over the two legs, staying on course for an unprecedented quadruple.
After a goalless first leg, the visitors took an early lead at the Emirates through Eden Hazard but Arsenal were level soon afterwards via a bizarre Antonio Rudiger own goal and scored the winner in the second half through Granit Xhaka.
Guardiola on the other hand will be desperate to claim the first silverware of his time in charge at City, who are chasing glory on four fronts.
Wenger was unable to field the cup-tied Henrikh Mkhitaryan, who joined the club from Manchester United earlier this week in a swap deal with Sanchez moving in the other direction while Ross Barkley made his debut for Chelsea as a first-half substitute.
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"We were not playing in the right position but we amended that in the second half and took control of the game."
- Sanchez 'great player' -
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"We lost a great player in Alexis Sanchez but when a team doesn't know what's going on in the dressing room there is less clarity and focus on performances," he said. "We know we have to deal with it now."
Antonio Conte's Chelsea, missing Alvaro Morata and Cesc Fabregas, had the better of the first half and had already had a goal ruled out for offside by the time Hazard opened the scoring in the seventh minute after Pedro released him through the middle.
Willian was forced off before half an hour and Conte threw on Barkley for his Chelsea debut but the former Everton man struggled to make an impact on the game.
A rejuvenated Arsenal started the second period on the front foot and edged in front in the two-legged tie when Xhaka reacted quickly to prod home a deflected Alexandre Lacazette cross with about half an hour to go.
But Conte's men, lacking a cutting edge, could not find the inspiration they needed to take the game into extra-time and Arsenal held on to claim a deserved victory.
It was the fifth time the clubs had met this season, with all four matches ending in draws, including 0-0 in the first leg of this tie.
The win is a major boost for Wenger in a week when the club, with a battle on their hands to qualify for the Champions League, lost Sanchez, arguably their most devastating attacking player.
"My thoughts are disappointment for the final result," said Conte. "I think my players must be pleased with their effort, we tried to reach the final, we conceded two goals and we were unlucky, two deflections.
"It is a pity because we had a good run in this competition, it is a pity for me, the players and fans," added the Italian.
The final takes place at Wembley on February 25.