Whites looked set for another early exit from the cup after a dazzling goal scoring from Mallik Wilks put Hull in the lead.
But as the game draws to a close, Ezgjan Alioski evened the scores in added time to force a penalty shootout.
Both teams were clinical from the start, but in sudden death, Jamie Shackleton’s saved effort proved crucial as the Tigers advanced to the third round.
Speaking in a brief press conference after the match, Bielsa said: “Without a doubt, they have put in enough effort. We didn’t defend well and we didn’t attack well.
“There was an error in the coordination between the three sectors of the team. The players did not stick together very well.
“It’s the coach’s responsibility and therefore I feel responsible.”
Former Leeds striker Mallik Wilks has wreaked havoc throughout the competition but Bielsa has admitted there is little his side can do to stop such a hot player.
“I don’t know if there is a direct recipe for stopping a player’s performance. Added the Argentinian.
MATCH REPORT: Ex-Leeds forward shines as Hull sinks Whites in epic penalty shootout