"Piku", also starring Deepika Padukone in the titular role, is about her relationship with her father (Bachchan), who spends most of his time obsessing over age-related issues.
Irrfan forms the third wheel of the story.
Shoojit now feels the megastar's chemistry with Irrfan is one of the highlights of the movie.
"I tell everyone Irrfan is the only international star we have. I was sceptical the first day he and Mr Bachchan met on the set, it was a Robert De Niro and Al Pacino moment for me.
"But it all went well... Apart from the pairing of Deepika and Irrfan, it is also a delight to watch Irrfan and Mr Bachchan together," Shoojit told PTI.
The director shared that by the end of the film, Irrfan became an admirer of Bachchan's style of working.
For Shoojit, "Piku" became a one-of-its-kind experience as the film's set was home to three different schools of acting.
"All three of them are absolutely poles apart. Mr Bachchan will constantly do rehearsals and then he will go for a take and that will be the final take. Deepika will do rehearsals but while doing that she will try to grab her movements, performances.
"Irrfan is very different. When he is acting and when not, is difficult to tell. It was great fun for me to have three different styles of acting. Irrfan said if somebody wants to learn acting and what is the process for it, they should work with Mr Bachchan. He was completely mesmerised by Big B's style of acting," Shoojit said.
Shooting the movie was a personal journey for the "Vicky Donor" helmer because the characters in the movie are inspired from people in his life.
"The idea for the film came from Juhi Chaturvedi, the writer. This story is a bit inspired by her grandfather. Then we brought in our expertise, like I brought a lot of sequences from my family...
"It has been a personal journey for me and it is going to be that for the people too because the characters are identifiable. Anyone can be Piku and any old man can be like Amitabh Bachchan's character," Shoojit said.
The director said the special feature of "Piku" is that it looks at everyday problems in a light-hearted way.
"Besides the relationship between the three characters, what interested me in the story is how in our family we have similar discussions about different events as in the movie and you think they are big problems but they are not.
"When you look back at these events, you laugh at them and smile. It is like you have are in a cosmopolitan city and you have problems to deal with but you still find a way to live life and how small relationships make your life fulfilled," he said.
"Piku" is slated to release on May 8.