The Delhi Belly star plays a Pakistani woman called Safar Khan whose son is accused of committing a murder PTI reports.
Jagannathan said of her role in the crime drama: "Safar Khan is the polar opposite of my character in Delhi Belly. I've never played someone like her, but the writing is so strong that it makes playing her feel like second nature.
"Her journey is a roller coaster. She goes from knowing her son is innocent to eventually thinking he might be guilty."
Co-written by Schindler's List writer Steven Zaillian, the series features an A-list cast led by The Sopranos star James Gandolfini, British-Asian actor Riz Ahmed, Bill Camp and Peyman Moadi.
Criminal Justice has won BAFTA Awards for best drama and best writer, three Royal Television Society Awards and an Emmy.
Speaking about the team behind the series, the 39-year-old actress said: "This is one of the most incredible projects I've ever been a part of. Everyone involved is [the] 'best in class'."
Jagannathan made her film debut in the Aamir Khan-produced Delhi Belly, and starred alongside Jane Fonda in Peace Love & Misunderstanding earlier this year.
The actress also stars in comedy drama Thanks For Sharing with Tim Robbins, Gwyneth Paltrow, Mark Ruffalo, Josh Gad and Joely Richardson.