A corner sailed over Sanchez in the seventh minute, allowing Ben White to head into an unguarded net, before the Spanish keeper let a low cross squirm through his grasp to gift Viktor Gyokeres a tap-in in the 49th minute at Stamford Bridge on Wednesday (Thursday AEDT).
Substitute Alejandro Garnacho pulled a goal back, but Martin Zubimendi restored Arsenal's two-goal cushion by slotting home in the 70th minute.
Garnacho's second goal in the 83rd minute gave Chelsea a lifeline ahead of the second leg on February 3 at Arsenal's Emirates Stadium.
Still, Mikel Arteta's team are favourites to advance to the March 22 final at Wembley Stadium in what is shaping up to be a banner season for the Gunners, who lead the Premier League by six points and top the Champions League standings.
Liam Rosenior, taking charge of his first home match as Chelsea's new coach, refused to criticise Sanchez for his mistakes.
"I said to him before the game and I made this really clear, when my players make mistakes, I'm accountable. That's on me," Rosenior said.
"I'll stay positive with him, as I will all the group. My job is to help them, not to blame them."
In the other semi-final, Manchester City won 2-0 at Newcastle in the first leg on Tuesday (Wednesday AEDT). Their second match is on February 4.
It means the most likely title match is City versus Arsenal, the top two in the league. That would be a rematch of the 2018 final - the last one Arsenal reached in the League Cup - which City won 3-0.
Rosenior was hired last week on a six-year deal and won in the FA Cup over the weekend.
That was away, against second-division opponents Charlton Athletic, so this was Rosenior's first real test, and Arsenal could have left Stamford Bridge in a commanding position. Instead, Garnacho's late goal meant Chelsea still have hope.
"We let them give us more pressure and that's how we conceded the goal. Not up to our standards really," White said.
Rosenior insisted on Monday that Arsenal shouldn't be labelled as "Set Piece FC" - a reference to the team's prowess in dead-ball situations - but the opener came from that route.
Sanchez came for the ball that was swung in by Declan Rice but it flew over him, while Chelsea striker Joao Pedro inexplicably failed to jump and attempt to clear. It left White with a simple task to head home.
That was Arsenal's 24th goal from a set piece in all competitions this season, according to the BBC, with 18 coming from corners.
It was a sloppy mistake by Sanchez, but his second was worse.
White's ball across the face of the six-yard box should easily have been gathered by the goalkeeper, but it went through his arms as he dived. Gyokeres couldn't miss for just his second goal since November 1.Â
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