The Bees looked set to make the last eight after taking the lead through Kevin Schade and managing a one-sided first half.
But Championship Wednesday equalized just before the hour mark with their first effort on target through Jaydy Gassama’s spectacular strike.
Brentford failed to find a winner in normal time but Dutch goalkeeper Flecken saved Liam Palmer’s penalty to send the home side 5-4 up after a 1-1 draw.
Thomas Frank clearly envisions a place in this competition and with at least five Premier League teams to be knocked out on Wednesday night, Brentford could further improve their path to the semi-finals in 2021 and perhaps even grab the first leg of the big silver tournament in the history of the club.
In contrast, Wednesday boss Danny Roll’s priorities are in the Championship as he made nine changes but they still gave Brentford a scare.
Brentford’s speedy starters duly went in front after 10 minutes when Keane Lewis-Potter’s pinpoint overhead kick found Schade who outran defender Gabriel Otegbayo towards goal.
Finishing has not been the German’s forte since he joined the Bees in January 2023 and Owls keeper Pierce Charles stepped up to block his initial attempt.
However, the ball ricocheted into Charles and hit Otegbayo to leave Schade with a simple tap-in for his first goal of the season and just his third for the club.
Charles was on hand to prevent a second 10 minutes later when Brian Mbeumo’s cross found Lewis-Potter whose effort curled and almost slipped under the crossbar.
Wednesday barely registered a shot of anger, but out of nowhere they equalized 12 minutes into the second half.
Flecken threw the ball to a galloping Schade near the center circle only for the striker to lose his footing.
Gassama picked up the loose ball, stepped forward and, with no one to close him down, fired a low rocket into the corner from 25 yards.
Brentford came back with Mbeumo forcing a great save from Charles from long range and Schade just couldn’t get a touch on Jaden Megoma’s through ball across goal.
However, neither side could force a winner and after nine successful spot kicks, Flekken dived down the right to beat Palmer’s effort and send the Bees ahead.