Brentford reached the semi-finals of the EFL Cup in 2020-21.
Brentford needed penalties to progress to the quarter-finals of the Carabao Cup on Tuesday following a 1-1 draw with Sheffield Wednesday.
The Premier League side dominated much of the tie and took an early lead through Kevin Schade, albeit in fortuitous circumstances.
The German striker's initial effort was saved by Pierce Charles but the ball rebounded back into his path off Gabriel Otegbayo and he was able to tap into an empty goal.
Brentford continued to get behind Wednesday's high defensive line and 19-year-old Charles was called into action to deny Keane Lewis-Potter and Bryan Mbeumo.
A much-changed Wednesday team managed just one touch inside the Brentford penalty area in the first half, but shortly after the break Djeidi Gassama drew Danny Rohl's side level with a superb strike from 25 yards out.
That goal sparked the match into life, with Ike Ugbo and Liam Palmer testing Flekken in quick succession before Mbeumo stung the palms of Charles.
Both sides pushed for the winner and introduced regular starters from the bench, but neither could find the breakthrough needed to avoid a penalty shootout.
The first nine penalties of the shootout were scored before Mark Flekken saved from Palmer to clinch victory for Brentford.
Brentford have reached the last eight for the first time since 2021-22, and will look to replicate their run to the semi-finals in 2020-21.
The draw for the quarter-finals will be made following the conclusion of the last-16 meeting between Tottenham Hotspur and Manchester City on Wednesday.
Quarter-final matches are scheduled to be played in the week commencing 16 December.
'I need to learn from that'
Flekken was the shootout hero, but he was not happy with the role he played in Wednesday's equaliser.
The goalkeeper tried to launch a Brentford counter attack after catching a corner, but Gassama intercepted his throw and scored.
"At the moment I caught the corner kick I had so many possibilities but I took the wrong one," Flekken told Sky Sports.
"The mistake I made helpd a lot to bring our mental side down. It took away the control we had. It went from a controlled game to an open game. I need to learn from that.
"I went to the wrong side twice at the start [of the shootout]. That doesn't help your confidence in a shootout but luckily I saved the last one.
"It's not the first and only mistake I’ve made in my career. I’m 31 now with quite some experience, that helps a lot [to come back from] a mistake like that."
With five Premier League sides guaranteed to be eliminated on Wednesday, Brentford manager Thomas Frank says his side are aiming to win the competition.
"It is one step closer to what we want to do, we want to go all the way," Frank told Sky Sports.
"Our mentality was great, the players worked hard. It was more [an issue with] quality."
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