Arsenal keeper Bernd Leno made two consecutive saves to stop Jota and Marko Grujic and the visitors responded with a chance of their own as Rob Holding’s header was punched away by Adrian in the 70th minute.
“It was a very tough game,” Leno told broadcaster Sky Sports. “It is very good to save some penalties and see some laughing faces but I prefer next time to win the game before penalties. But the main thing is we won.”
The Gunners also appealed for a penalty in stoppage time when James Milner fell and handled the ball.
The referee Kevin Friend did not point to the spot and there is no VAR in the competition.
Arsenal, however, celebrated in shootout when Joe Willock’s spot kick crept under keeper Adrian to send them through.
Meanwhile, a first-half header from Sam Vokes gave second-tier side Stoke a 1-0 win at Aston Villa to set up a quarter-final at home against Tottenham.
Villa manager Dean Smith made 11 changes from the team which beat Fulham 3-0 on Monday.
Fulham lost 3-0 again — this time against second-tier side Brentford.
Scott Parker’s side has also lost all three Premier League games since gaining promotion in August, when it beat west London rival Brentford in the playoff final.
Algerian midfielder Said Benrahma scored twice after Finnish forward Marcus Forss’ opener as Brentford reached the League Cup quarterfinals for the first time. It will play at home against another Premier League side, Newcastle.
Manchester United will travel to Everton in the other quarter-final.