Jessup began acting at the age of 11 as a child actor. After various early jobs, including a role in a stage adaptation of The Full Monty, Jessup got a lead role in the children's television series The Saddle Club, for which he is also credited with conceptualizing the plot of one episode. He served as executive producer for the independent film Amy George, which played at the 2011 Toronto International Film Festival.
In 2011, Jessup was cast as a main character in the TNT science fiction series Falling Skies. He starred on the show for five seasons. In 2012, Jessup starred in the Canadian indie film Blackbird, in which he played a troubled teen falsely accused of planning a school shooting. Jessup's performance received positive reviews and the film won various awards, including the Best Canadian First Feature Film award at the 2012 Toronto International Film Festival. The film won three awards at the Atlantic Film Festival and won the Vancouver International Film Festival award for Best Canadian Feature Film. It also screened at the Cannes Film Festival.
In 2015, Jessup also starred in the feature film Closet Monster, which won the Best Canadian Film Award at the 2015 Toronto International Film Festival, and screened at various film festivals around the world, before subsequently being released nationwide in July 2016.
In 2016, Jessup starred in the Emmy-winning ABC series American Crime as Taylor Blaine. His performance has been praised by critics. He returned for the third season, as Coy Henson.
Jessup has stated in numerous interviews his intention to work in the film industry beyond acting, as a director and filmmaker. In an interview with the Toronto Star, Jessup said that directing “is where my mind and heart [are] at the moment". In 2014, Jessup stared in short film Fragments, and in 2015, Jessup's short film Boy, which was funded by bravoFACT, which he wrote and directed, premiered at the 2015 Toronto International Film Festival.
In 2016, Jessup was included in Filmmaker Magazine's annual list of the 25 New Faces of Independent Film. In August 2016, his short film Lira's Forest, which he will be writing and directing, received funding from the Harold Greenberg Fund's Shorts to Features program, with Albert Shin as his executive producer. The film premiered at the 2017 Toronto International Film Festival. Jessup wants to expand this story with a feature film, Simon's Forest.
As a filmmaker, Jessup has expressed a particular fascination with Japanese film and culture and has stated in interviews that he is highly influenced by East Asian cinema, including directors Hirokazu Kore-eda, Edward Yang, Yasujirō Ozu, Hayao Miyazaki, Isao Takahata, and Apichatpong Weerasethakul. He has also expressed admiration for directors Abbas Kiarostami, Andrew Haigh, and Ira Sachs.