The Liverpool goalkeeper thinks his side have a good chance of FA Cup glory.
However, only Liverpool, Manchester United and Arsenal remain in the tournament of the Premier League's top seven sides, with Chelsea, Manchester City and Southampton having all exited the completion last time around.
And the Belgium international feels his side have a real opportunity to win some silverware under Brendan Rodgers, who is yet to win anything as Liverpool manager.
"When you are in a cup competition, you just want to go as far as possible,” said Mignolet.
“We showed great character after the penalty to come back and score twice in the last five minutes. It is a big relief, especially when you can do it in the 90 minutes rather than go to extra time - especially with the derby (on Saturday) in mind.”
Philippe Coutinho’s stoppage-time strike secured the dramatic 2-1 win over Neil Lennon’s side, after Raheem Sterling had initially drawn the sides level following Eidur Gudjohnsen’s penalty.
The Trotters had to play the last 25 minutes with 10 men following Neil Danns’ dismissal and eventually buckled under the pressure, though not before giving Liverpool a real scare. Coutinho’s late strike saved the day, though, and Mignolet is hoping it will give his side the confidence to go on and win the tournament.
"It can only help the confidence. The main thing is to progress in the cup because we have a big chance,” he added.
"It does not matter who you come across. Of course we have the belief we can do it - at Liverpool you want to win every competition you feature in, not just the FA Cup."